Progress towards God comes through the mercy of the spiritual master. A pure devotee can confer upon a receptive disciple the greatest gift of love for God.

Abandon unclean, polluted thoughts from your heart. Fill your heart with compassion and love.

There is no higher emotion than love. There is no higher perfection than love.

The mind can be your dearest friend. The mind can be your greatest enemy. O Lord! Bestow Your mercy upon me so that I reach You cutting through all intervening layers.

Regulate your life. Through the fire of regulation burn to ashes millions and millions of unclean thoughts and desires.

Abandon bad association as a plague. May your consciousness ever be fixed firmly on the eternal truth.

May the Supreme Lord fill the seeker with knowledge. May your heart be filled with love.

O Lord! Make Thy presence felt at all moments in my life! O Lord! Who but You can give hope to the hopeless! Who, but You, of infinite love, can mend the heart of one separated from You! Who but You can rescue this most fallen soul from the miseries of material existence!

O Lord! Invisible to these eyes of mine, You break my heart to pieces! I cry out to You, O Lord, just as a helpless child cries out to its mother. I know nothing but You, and yet I suffer forgetting You. Fully surrendered to You, I plead, O Lord, take control of me. You can make this servant of Yours dance as You like. Engage me in Your service, O Lord!

O seeker of the Truth! Worry not! Fear not! Despair not! Fill yourself with all virtuous qualities. Let your life itself be an offering to God.

That which is yours, O friend, is eternally yours. That which is not, was never yours. Misidentifying and falling into delusion, you have forgotten yourself, O brother.

May that Supreme Lord of all the universes bestow His mercy upon us so that we may progress towards Him.

How wretched I must be
Feel I do not even an iota
of love for that most loving One
The all-attractive, beautiful and sweet Krsna.

Taking solace in hearing about Him
I shamelessly manage to live in separation
When will that day be mine
when tears of separation cover my face.

O dear beloved! You certainly are cunning
You tease, and then abandon
Is this how you treat your loved ones,
O maintainer of religious principles.

But you still remain unconditionally
My master for all time to come
O Lord of Vrndavana, be merciful
upon this servant of Yours,
and favor me thus, O Lord,
with eternal love for You.

.

Lord Krishna instructs His dear friend and devotee, Arjuna:-

When you remember Krishna in all activities, at all times, your whole consciousness will become purified.

Pray that Krishna bestows His causeless mercy upon us so that we may remember Him, and go back home, back to Godhead.

Pray sincerely to God. Read about the stories of God. O how beautiful are the stories of Rama and Krishna. Read them. You will certainly be blessed. There is no doubt regarding this. Know that God is with you at all stages and under all circumstances. Worry not, dear child. You are the child of God. He, Who takes care of you, Who has taken care of you since beginningless time, and Who will always take care of you in the future, is taking care of all those around you. So, do not worry about others, O dear child. There is nothing to worry for a devotee of God.

Simply chant the glories of God, O child. Surely, you will be led to the correct path. Every living entity has to experience the reactions to actions conducted for millions of births. However, the devotee of God is not sad at his own fate, nor is he unwisely sympathetic seeing external suffering in others. He knows that whatever happens, happens according to the will of God. He prays to God, “O my dear Lord, please grant me this benediction – whatever condition of life I may be placed in my next birth – be it hellish or heavenly or anything else, and whatever else I may experience in this birth, may I always remember You. May I never forget You. May I always know that You are with me.” Thus, a devotee has no fear, since he always has God in his heart. The devotee is infinitely compassionate. He always prays that great love for God may arise in the hearts of all.

O great are the glories of one who has taken the name of God even once! So, rejoice, O child, since you are blessed. Be happy, not morose. Krishna says in the Bhagavad Gita – “the transcendentalist is ever cheerful at heart”.

May Narayana always be with you. Hare Krishna.

There is no fear for me now,
O only fearlessness do I know,
It is in fearlessness that I breathe,
In fearlessness that I live.

All fear is gone,
No traces left behind,
Greater fears may come,
But they too would be handled.

Whatever comes shall be dealt with,
The son of Devaki and Vasudev shall help,
How can any fear come near
to one who takes shelter in Him.

He, Krishna, the destroyer of fears, is real,
So, have no fear, O brother,
Take refuge in His holy name,
And you will be delivered from the greatest of fears.

Just chant His holy name,
Have immense faith in Him,
Assuredly He will protect you,
And assuredly He will guide you.

The friend of Arjuna,
The cousin of Uddhava,
The son of Devaki and Vasudev,
The son of Yashoda and Nanda,
The lifter of the Govardhan hill,
The killer of the demon Putana,
The killer of the evil Kamsa,
The friend of the Pandavas,
The protector of Draupadi,
The brother of Balarama,
The nephew of Kunti,
The lord of Bhishma,
The lord of Radha,
The friend of all,
The lord of all,
The avatara of Vishnu,
The brother of Durga,
The Supreme Lord of all.

You are taken care of. Everyone is taken care of.

The eternal all-courageous power is within you. Fear will not touch you when you realize God. Be like a lion, O great warrior. Not trembling at noises, in the dead of the night, roaming majestically in the fearsome jungle. Arise, O child, and abandon fear.

Sacrifice your actions, thoughts and all to God. Sacrifice your past, present and future to God. Work towards perfection, and towards love, and towards sincerity, and towards fearlessness. Love all. Serve all.

There is nothing but Allah. He who is Krishna is Allah and Jesus.

He came and played in Vrindavan as Krishna, and in Bethlehem he came as Joseph’s son. He came once as the child of Devaki and once as the child of Mary. Blessed are you, O Mother Mary, as are you, O Mother Yashoda.

Follow what Lord Jesus said – “Love your neighbor. Love all as your own. The Kingdom of God is within you. Become like a child, and you will find the Kingdom of God. You are the blessed children of God, my children. The Lord God is here. Be meek and humble, and you will reach Him.”

He is not reached by arrogant scholarship. He is not reached by stone-hearted selfishness. He is not reached by hatred.

May you be filled with devotion,
May your heart be filled with love,
May you become love itself,
May your being be one of love.

Love all as your brothers, my friend,
Love all as your own,
See beauty in all, my friend,
May your heart be filled with love.

Take the name of the most beautiful Krishna,
Of the most merciful one,
Of the most compassionate,
Of the most loving one.

Dance in the bliss of Krishna’s name,
Sing, rejoice, my friend,
May your heart be filled with devotion,
May you beceome the embodiment of love.

Ask Krishna for his mercy,
Not just for you, but for all,
Ask that the suffering of all may come to you,
When you ask so, you will get bliss.

Serve the poor and the weak, my friend,
See Krishna in all of them,
Be most humble, my friend,
May your heart be filled with love.

Hari bol, Hari bol! Nitai Gaura Hari bol!

Hare Krishna Hare Krishna Krishna Krishna Hare Hare
Hare Rama Hare Rama Rama Rama Hare Hare

A few months later, Hodja was back at the hotel. This time, the same waiter gave him incredibly good service. Before leaving, Hodja left the tiniest copper coin as a tip. The waiter was again stunned. He asked Hodja, “Brother, explain to me your behavior.” And Hodja replied, “The gold coins were for today’s service. The copper coin is for the service you had given last time.”

And at last Hodja came. He did not what to speak on. He asked the crowd – “Do you know what I am going to tell you?” And they replied, “No, dear Hodja, we don’t.” Hodja snapped, “In that case, I am not going to tell you anything.” Saying this, he left.

The next week, again a huge crowd assembled in expectation of Hodja’s sermon. Hodja came in due time, and asked, “Do you know what I am going to say?” The crowd had learned their lesson from the previous sermon, and replied in unison, “Yes, we do know what you shall speak.” And Hodja said in great joy, “In that case, I shall not speak anything.” Saying this, he left.

Another week passed, and it was again the day of the sermon. The crowd was prepared this time to do whatever it took to make Hodja talk. Hodja arrived, and asked, “Do you know what I am going to say?” As was planned, some members in the crowd replied “Yes” while the others said “No”. And Hodja said, “In that case, those of you who know can speak to those that don’t.” Saying this, he left.

Unselfishness is good to cultivate. Selfishness leads to suffering. Unselfishness might be harder to practice than selfishness, given that selfishness dominates many of our actions. However, if we try consciously, we can make unselfishness the basis of many of our actions. Assuredly, you would find great courage and strength and clarity through constant practice of unselfishness.

What did all the saints tell us – do good, and be good. Orient your life in that direction. Without doubt, you would find yourself to be an instrument of God, and His grace will shower upon you.

However, do not despair at any wrong actions that you happen to commit. Though we would wish often to lead our life doing good and being good, there would be times when that does not happen. So you happen to commit something that you would not consider good. Do not worry; do not regret. Strive always to improve; strive always to root out the selfishness in you. Strive so that the suffering caused by wrong action is converted to courage and strength.

Celebrate every moment. Life is indeed the greatest celebration. Rejoice so that your brother may rejoice. Rejoice because your brother rejoices. Rejoice, laugh, be good, have no fear – you would certainly progress.

O dear friend, the teacher is within, and outside. The Teacher is the embodiment of the mother’s love, and the father’s love. Work, in whatever station of life you are – if you are a son, serve your parents; if you are a student, study; if you work, work well. Know that the grace of the Lord is ever with you. Have no fear. Sacrifice the results of all actions to God, and you will reach God. Have no fear or regret of past, present or future – work, pray in the knowledge that you shall realize God one day.

Say the name of Rama, my friend,
Dance in joy, rejoice in Rama,
Say the name of Rama, my friend,
And assuredly he will protect you,
He is the courage in you,
He is the strength in you,
He is knowledge in you,
He is the warrior in you,
Taking his name, you shall never fall,
No misery shall affect you.
Take the name of Mother Sita,
Take the name of Bhakta Hanuman.

Make me Your instrument, O Mother,
And through me, may You act.
May I bring Your love to this world.
May I bring joy to this world.
May the sufferings of all cease.
May bliss fill the hearts of all.

Give me Your rod, so that others may experience bliss,
Bring on me the sufferings of all, O Mother,
So that the sufferings of others may ease.

Sincerity, my child. That is all you need. I saw the girl with sincerity written all over her being, and in her saw the sincerity of God. I saw the girl with perseverence and concentration written all over her being, and in that saw the face of God.

Unselfishness is what you need. Live not a moment mired in the cesspool of selfishness. Live for others. Unselfishness is the only virtue that should guide you as you live in the world. Take on pain, take on suffering. And with guarantee, you would find the Kingdom of God. Love thy neighbor, and live for thy neighbor, and you will find that God Himself is in all.

Abandon all fears, and all notions of guilt. So what if have done wrong in the past? So what if you are doing wrong in the present? So what if you do wrong in the future? Fear not, my dear sister, you will transcend life that you know to an eternal life. Strive to be unselfish. And yet, if you do act selfishly, and happen to do harm to others, do not despair. Strive, strive, strive until you find the hand of God on your head, until you find the arms of God lifting you and carrying you. At that time, you would be an instrument of God, and your actions would be a manifestation of God’s will. That stage may be far; that stage may be near. But what of that? Fear not, and continue to struggle ahead like a warrior. Despair not even at the greatest of miseries. It shall not only pass, but shall verily be transcended.

Fill yourself with strength, dear sister and brother. Inner strength will help you in the practice of unselfishness, and unselfishness would by itself build your inner strength. Fear not, fear not! Courage is the only refuge for the warrior.

Unselfishness without courage or strength is not good.

Fear not, my child. You are looked after. He, Who created the Universe; He, by Whose will the sun gives light, and moon gives coolness; He, Who has looked after you since beginless time, know that He is always looking after you.  

All that has happened, all that is happening, and all that shall happen  – fear not, regret not, despair not at any of these. Taking refuge in unselfishness, strength, courage and faith in God, know that you shall reach eternal life.

What glimpse did You give me,
just to take away the next moment
Still in that moment, I stay
Wanting to get back to that.

Now, where did You go, O Lord?
You, whom I saw? You, whom I felt?
Wandering in the city streets I shout out loud,
O Lord, O Lord, O Lord!

Love are You, Truth are You,
May all glories be to You

Worship the Lord, my friend,
He is real, He is the supreme,
He creates the universe,
He creates the sun,
He creates the moon and earth,
May all glories be to Him.

But Hodja could not reply. Laughter was always inside and outside. When he laughed, everyone laughed.

Then he met his friend, Bhoja, and Bhoja said – “my dear Hodja, you should cry, because sad and bad things have happened to you.”

That very moment, Hodja started wailing, “Waaaaaaaaaaaaaaa”.

Passersby saw. One came and said, “why do you wail, dear Hodja?”

And Hodja said, “I was told that what happened was sad, and what happened was bad. And was told to be sad. And so I cry.”

Be one with Yourself. And lo, what is the meaning of fear.

What did our beloved Jesus say? Love thy neighbor. That was the call. That was what was said. Why then do you strive to hate your brother?

Love is God. Let the love of God flow through you, as it did in our dear saints Francis and Chaitanya. Verily become an instrument of the mercy and peace of the Lord, and finding love within, you would see nothing but love outside.

Thus, annihilate yourself. Become love personified. Eschew hatred. Liberation will be yours for the taking. You would have conquered all that there is.

What is there for me to do,
When the Universe is all that exists.
When what is real and unreal is not known.
O Lord! Make me an instrument of Thy mercy,
And annihilating me, act through me.

O Lord! Am I one with You?
Am I distinct? Do I know You?
Or do I not?
Ah the confusion and the reality.

I know nothing but You,
see and hear nothing but You,
Sacrificing everything at Your feet,
When life and death are transcended,
O Lord, you will be reached.

I ask of Thee, O Father of  the Universe,
Fill me with unselfishness.
May unselfishness fill every pore
May unselfishness guide every thought
Seeking refuge in unselfishness,
May I proceed to the Real.

O Lord! Forgetting You, I say
Fill all with knowledge.
Fill all with unselfishness.
Fill all with fearlessness.

O seeker! Strive always to proceed to knowledge, fearlessness, strength, unselfishness and an overpowering power of sacrifice, and a willingness to take on sufferings of the universe upon yourself. Do not sacrifice rationality at the altar of selfishness. May we struggle towards an intellect like that of Adi Shankara and a heart like that of the Buddha.

Work hard, and struggle hard. Sincerity is always rewarded.

Arise, since you have slept for long! Arise, since realization has dawned upon you! Arise, since you can no longer relax!

Arise, since your Mother needs you! Arise, since you need your Mother! Arise, O dear brothers and sisters of mine, since it is already noon, and the sun is about to set!

Work, since through work you will reach God. Work, since through work you will pray. Work, since through work you will meditate.

Work, since your brothers and sisters and sons and daughters and mothers and fathers are suffering. Work, so as to lessen the suffering of your neighbor. Work, so as to make yourself an instrument of the grace of the Lord.

Work, since there is physical suffering in the universe. Work, since there is mental suffering in the universe. Sleep no longer. Dream no longer. The time is for action. The time is to wage the greatest of all wars. The time has come to conquer your self.

Work, knowing that the work is not done by you. Work, knowing that the fruits of the work are not for you. It is the Universe that works through you, and it is the Universe to whom the fruits of the work belong. Hence, abandon all fear of action – past, present and future.

Cultivate unselfishness. Unselfishness is to take on suffering willingly so as to lessen the suffering of others or so as to increase the happiness of others, however small and temporary the lessening of suffering or increase in happiness may be. Throw aside all selfishness. Live life not for your own limited self, but for the Universe. Then, and only then, would you feel the current of destiny flowing through you, and you would verily become destiny’s child.

Have you ever thought for a moment or meditated for a moment on the suffering of your own brother or sister or mother or father or husband or wife or son or daughter? Have you ever experienced the suffering of any one in your heart? What has selfishness gained you but untold pain and suffering? Cast it aside in disgust. Fill your heart and mind with the suffering of your mother and father and brother and sister and husband and wife and son and daughter and neighbor.

What have you gained by making your heart as hard as stone? You say that you gained a self-assurance, a mental poise, and strength through that. I say that this self-assurance and mental poise and strength are but waiting to disappear. It is true that one whose heart is like a stone feels no pain, experiences no pangs of suffering. But then, he also experiences none of the human qualities of love and compassion. He becomes verily a stone – inert and dead. Therefore, resolve to cast off this stone-heartedness, and make your heart tender and loving. No salvation is possible for one who remains stone-hearted.

Cry, cry, cry – not one moment for yourself, but always for the ones around you. When you experience the suffering of the universe in your heart, how will tears not flow? how will your heart not cry out? Cry, because through these tears will arise purity. Through these tears will arise strength. Through these tears will arise fearlessness.

Enough of the so-called inner strength which is built upon the foundation of fear. Enough of the so-called inner strength which makes a man an island unto himself. Enough of the so-called inner strength which leads not to spiritual progress but to destruction.

Inner strength should be based on unselfishness. Inner strength should be based on courage. Inner strength should be based on an infinite tenderness of heart. Be like a mother to her child – in her complete surrender to the child, the mother is filled with infinite strength.

Throw out all thoughts of weakness. Remember always that weakness is the greatest sin. Fear is the greatest sin. Always meditate on your true infinite self. Know that you are the reservoir of all courage and fearlessness.

Hatred is a great enemy. Why do you hate another just because he does things differently from you? Why do you judge another and pronounce condemnations on him. Know for sure that hatred will lead you astray. For millions of years have you filled your heart with hatred that it has forgotten how to love.

Recall to mind the immense amount of work that needs to be done. Recall to your mind the suffering of all beings. Recall to your mind the great mercy and grace shown to you in all stations of your life. Recalling thus, abandon once and for all, all weakness and fear. Arise with strength, O great warrior! The time has come to awaken! Thou shalt not be found wanting!

Courage, unselfishness, immense faith,  indomitable spirit, willingness to suffer, unceasing work – cultivate each of these qualities.

Make me a channel of your peace:

Where there is hatred, let me bring your love,

Where there is injury, your pardon, Lord,

And where there’s doubt true faith in you.

Make me a channel of your peace:

Where there’s despair in life, let me bring hope,

Where there is darkness, only light,

And where there’s sadness, ever joy.

O Master, grant that I may never seek

So much to be consoled as to console;

To be understood as to understand,

To be loved, as to love with all my soul!

Make me a channel of your peace:

It is in pardoning that we are pardoned,

In giving of ourselves that we receive,

And in dying that we are born to eternal life.

Here is the beautiful song being sung in a beautiful voice:

Here is a translation taken from http://harekrishnatemple.com/siksastakam.html.

1. Glory to the Sri-Krsna-Sankirtana, which cleanses the heart of all the dust accumulated for years and extinguishes the fire of conditional life, of repeated birth and death. This sankirtana movement is the prime benediction for humanity at large because it spreads the rays of the benediction moon. It is the life of all transcendental knowledge. It increases the ocean of transcendental bliss, and it enables us to fully taste the nectar for which we are always anxious.

2. O my Lord, Your holy name alone can render all benediction to living beings, and thus You have hundreds and millions of names, like Krsna and Govinda. In these transcendental names You have invested all Your transcendental energies. There are not even hard and fast rules for chanting these names. O my Lord, out of kindness You enable us to easily approach You by Your holy names, but I am so unfortunate that I have no attraction for them.

3. One should chant the holy name of the Lord in a humble state of mind, thinking oneself lower than the straw in the street; one should be more tolerant than a tree, devoid of all sense of false prestige, and should be ready to offer all respect to others. In such a state of mind one can chant the holy name of the Lord constantly.

4. O almighty Lord, I have no desire to accumulate wealth, nor do I desire beautiful women nor do I want any number of followers. I only want Your causeless devotional service, birth after birth.

5. O son of Maharaja Nanda (Krsna), I am Your eternal servitor, yet somehow or other I have fallen into the ocean of birth and death. Please pick me up from this ocean of death and place me as one of the atoms at Your lotus feet.

6. O my Lord, when will my eyes be decorated with tears of love flowing constantly when I chant Your holy name? When will my voice choke up, and when will the hairs of my body stand on end at the recitation of Your name?

7. O Govinda! Feeling Your separations I am considering a moment to be like twelve years or more. Tears are flowing from my eyes like torrents of rain, and I am feeling all vacant in the world in Your absence.

8. I know no one but Krsna as my Lord, and He shall remain so even if He handles me roughly by His embrace or makes me brokenhearted by not being present before me. He is completely free to do anything and everything, for He is always my worshipful Lord, unconditionally.

Many a lifetime have you wandered,
Many a chance have you squandered,
Given yet another chance to do right,
Drink the Bliss of Ecstasy tonight.

Drown yourself in the Universal Spirit,
Lose yourself in nectarean thoughts.
Hooking yourself onto the Soul
Soar to heights unknown to you.

What has travel gained you?
What gain is there in the Unreal?
Fix yourself, O Mind, on the One Real,
And go to depths unknown to you.

1. What is the difference between shallow copy and deep copy?

2. What are the functions C++ provides to our classes by default?

3. When should we provide our own copy constructor and copy assignment operator?

The posts have been taken from a file I had written for aiding in preparation for software engineering interviews. The style of exposition in the posts would hence exhibit a marked difference from that of a textbook, and orient itself more towards an informal synopsis written with the express purpose of aiding the understanding of the reader, who is assumed to be familiar with certain C++ concepts.

The pdf file which forms the basis for these two posts is attached here -  C-tutorials

We can approach this problem using a lookup table. Using a lookup table, the running time can be significantly reduced.

For simplicity, let us assume that our input string can only contain characters from the English alphabet i.e. only 26 characters are possible, though each may be repeated any number of times.

The idea is to construct a lookup table having space for 26 entries. Each entry in the lookup table corresponds to a character in the input string.

We will proceed as follows— Create a lookup table of size 26. Initialize each entry in the lookup table to be 0. The entries in the lookup table correspond to the 26 characters in the English alphabet, and are equal to the number of times that particular character appears in the input string.

Traverse the input string. Say the string that you are given is “elephant”. The first character is ‘e’. Go to the lookup table entry corresponding  to ‘e’ and increment the count in that entry by 1. But which is the entry in the lookup table corresponding to ‘e’?

Let us call the lookup table L.

L[0] will store the count of ‘a’. L[1] will store the count of ‘b’. … L[25] will store the count of ‘z’.

So, for ‘e’, we go to L[4] and increment that count by 1.

Now, once we have finished traversing the string, and have the counts of all characters available to us in L, we need to output the first non-repeated character.

For this, we proceed as follows:

Traverse the input string again. Take the first character in the input string. Go to the entry corresponding to that character in L. (Note that we can do this in O(1) time.) Check if the count is 1. If so, output it. Else, go to the next character in the input string, and repeat the same procedure.

————

Now, what would be the running time of this procedure?

Initializing the look up table takes O(1) [since the size of the lookup table is a constant.]

Traversing the string to populate the lookup table with appropriate counts takes O(n) time. (n is the length of the input string.)

Traversing the string again, to output the first non-repeated character takes O(n) time.

Hence, the total running time is O(n).

The memory requirement is that of a lookup table of constant size, which is O(1).

———

In C++, we can code it as follows:

——–

#include <iostream.h>

//func f returns the first non-repeated character in s.

char f ( char *s)

{

//check for null string.

if ( ! s)

return ”;

int L [26] = {0}; //lookup table, initially all entries are 0.

//we now traverse the string to populate L.

char *t = s;

while (  *t != ”)

{

L [ *t - 'a']++; // e.g. if *t is ‘e’, we need to increment L[4] which is L ['e' - 'a'].

t++;

}

//we have now populated L with counts.

//we now traverse s and using L, output the first non-repeated char.

t = s;

while ( *t != ”)

{

if ( L [ *t - 'a' ] == 1)

return *t;

}

//no non-repeated char in s.

return ”; //we may also write: return 0; (using the ASCII value of NUL)

}

———-

Now, what if remove the simplifying assumption made earlier and say  that s can contain any character. In this case, the lookup table needs to have 128 entries (one for each ASCII character).

For the preceding program, we use L [ ch - 'a' ] to access the lookup table entry corresponding to ch. In the new lookup table having 128 entries, we can use the ASCII values of the characters as direct indices into the lookup table, i.e. we can use L[ch] to access the count of character ch.

The remaining part of the code would remain more or less the same as above.

———-

The previous approach consider each character of the string and then compared it with each of the remaining characters (albeit only those following it) to check if it gets repeated anywhere. Hence it took time. We can do better than that by sorting the string. Once the string is sorted, checking if any given character in that string gets repeated becomes a simple matter of checking the elements adjacent to it.

What we propose to do is this:

- Copy the original string into a new string.

- Sort this new string. (This will take time. )

But now we are faced with a problem.

We need to determine a non-repeated character in the original or new string. If we wanted just any non-repeated character, that would have been simple. — We would traverse through the sorted new string, and in time, we would have been able to output a non-repeated character in that string. Hence, the total running time would have been . However, what we want is  the first non-repeated character.

One idea could be to proceed as follows: Take the first character in the original string. Using  the sorted string, check if that character is being repeated. If not, return that character. Else, go to the next character in the original string and repeat the same process again.

But again, we are faced with a problem — How do we determine the location of the first character of the original string in the new sorted string in constant time? (We can, of course, do so in O(n) time. But that would only lead to an O(n^2) algorithm, which would mean that even after sorting we weren’t able to improve the running time.)

—–

From the above, we observe 2 things –

1. Sorting would help in finding repeated / non-repeated characters in a string.

2. But sorting, as such, would not help us efficiently find the first non-repeated character, because we cannot perform a constant time lookup for any given character in the sorted string.

The solution is somewhat clear given the above observations. Do a sort akin to bucket sort. i.e. keep a bucket for each character that a string can possibly have. Once you have that array of buckets each corresponding to one possible character, you can perform constant time lookup in that array for any given character. i.e. we can construct a lookup table. This leads us to the approach detailed in the next post.

Now, how do we go about finding this first non-repeated character programmatically?

APPROACH 1

There is a very direct algorithm which we can employ — Take the first character of the given string (which is ‘a’ in our example string “alligator”). Compare it with each of the other characters in the string. If that character does not appear anywhere else, return that character and terminate the algorithm. Else, if the character does appear somewhere else in the string (as in the case of the ‘a’ in “alligator”), go to the 2nd character of the string, and repeat the same process (i.e. check if the 2nd character of the string gets repeated anywhere). Continue this process for each character of the string (i.e. in the k th iteration, we would compare the k th character of the string with each character in the string to determine if it appears anywhere else). If, at the end of the algorithm, we find that all characters get repeated somewhere, i.e. there is no non-repeating character, print a message to that effect.

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The above algorithm is simple to understand and implement. It will run in time where is the length of the input string.

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In pseudocode, the algorithm would look as follows:

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Input: String s of length n.

Output: First non-repeated character in s.

Algorithm:

For i going from 0 upto n-1:

flag = true; // flag is a boolean variable initialized to true.

For each j going from 0 upto n-1:

If ( s[i] == s[j] )

flag = false;

End For. //end the inner For loop.

If ( flag == true ) //i.e. if flag is true when we come out of the inner For loop

return s[i];  //character s[i] is the 1st non-repeated character in s.

End For. // end the outer For loop.

//If the algorithm reaches this statement, it means that there is no non-repeated character in s.

// Hence we return a default character, which would not otherwise be returned.

// We choose the null character — NUL for this purpose.

return NUL;

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So that was the pseudocode for the algorithm. We can translate the above code into C++ as follows:

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#include <iostream.h>

// function f takes a string s, and returns the first non-repeated character in s.

// In case there is no non-repeated character, it returns the null character ” (having ASCII value 0)

char f (char *s)

{

//we first check if s is the null string.

if ( !s)

return ”; // return the null character is s is the null string.

int i = 0, j;

bool flag;

while ( s[i] ! = ”)

{

flag = true;

j = 0;

while ( s[j] ! = ”)

{

if ( s[i] == s[j] )

flag = false;

j ++;

}

if (flag)

return s[i];

i++;

}

return ”; //i.e. no non-repeated character was found.

}

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Let’s say we need to find the binary representation of .

The method that most of us would have been taught in school is the following:

- Divide 100 by 2. You get 50, with remainder = 0. Write 0.

- Divide 50 by 2, to get 25 and remainder = 0. Write 0.

- Divide 25 by 2 to get 12 and remainder = 1. Write 1.

- Continue doing so until the number (which is 25 at present) becomes 0.

- The binary representation of 100 is the reverse of what you have written.

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We can write a recursive function to compute the binary representation of a decimal number. Using the power of recursion, we can avoid having to reverse the output as is necessitated by the above method.

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// Function dec_to_bin outputs the binary equivalent of the decimal number n.

void dec_to_bin ( int n)

{

if ( n < 0)

{

cout << “-”;

n = -n;

}

if (n == 0)

return;

dec_to_binary (n/2);

cout << n%2;

}

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dec_to_bin will take time proportional to the number of bits in the binary equivalent of n.

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The underlying method is the one discussed at the beginning of the previous post. At each iteration, we divide a number by 2 and store the corresponding remainder in an array. At the end of the process, we reverse the contents of the array.

Now, we are faced with a small problem. How do we know what the size of our array should be? Fortunately, we can proceed without knowing that, using the resizeable vector container class. (A vector is like a dynamic array. We need not specify its size at the time of definition.)

In order to reverse the vector, we can simply use the reverse algorithm defined in the algorithm header. Or if we simply want to output the binary representation (without needing to store it anywhere), we may simply use a reverse_iterator to iterate through our vector.

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// function dec_to_bin outputs the binary equivalent of the decimal input n.

#include <iostream>

#include <vector>

using namespace std;

void dec_to_bin ( int n )

{

vector <int> v; // vector of ints to store the binary representation.

int temp = n;

if ( n < 0)

temp = -n;

while ( temp != 0)

{

v.push_back (temp%2);

temp / = 2;

}

vector<int> reverse_iterator i;

if ( n < 0)

cout << ” – ” ;

for ( i = v.rend ( ); i != v.rbegin ( ); ++i)

cout << *i;

}

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If we used a list or a deque instead of a vector, we wouldn’t be faced with the task of having to reverse the values computed by us. We can do this by storing successive numbers that we generate at the front of the data structure instead of at the end.

We may do that using the push_front member function for lists and deques. (Note that we cannot use push_front for a vector. We can only insert and delete elements (efficiently i.e. in constant time) at the back of a vector.)

——

The above program using a deque along with push_front would look as follows:

——

#include <iostream>

#include <deque>

#include <algorithm>

using namespace std;

void dec_to_bin ( int n )

{

deque <int> v;

int temp = n;

if ( n < 0)

temp = -n;

while (temp != 0)

{

v.push_front ( temp%2);

temp /=2;

}

if ( n < 0)

cout << “-”;

copy (v.begin( ), v.end ( ), ostream_iterator<int> (cout,”");

}

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Instead of using an iterator to iterate through the deque, we can use the copy algorithm defined in the header algorithm to directly output the contents of the deque. (Note that the statement #include <algorithm> though written as part of the above program is not necessary, since algorithm.h automatically gets included when we write #include <deque>. )

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Firstly, we assume that the exponent is an integer.

For example. let us compute (13.5)^5.

Now, we know that 5 is 101 in binary notation.

Hence, what we need to compute is

We also observe that and that .

From these observations, we can deduce that repeated squaring of and multiplication of the appropriate intermediate results would lead us to the desired answer. (i.e. Compute and take the product of and .)

Here, prior to the repeated squaring, we calculated the binary representation of the given exponent. We need not necessarily do that. We can do the squaring and the computation of the binary representation simultaneously.

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Here is a C++ code fragment to illustrate that:-

//function f takes a computes x^e (where e >= 0)  and returns a long value.

long f (int x, int e)

{

int temp = e, rem; //rem stands for remainder.

long answer = 1, term = 1;

while ( temp != 0)

{

term = term * x;

rem = temp%2;

if (rem == 1)

{

answer = answer * term;

}

temp / = 2;

}

return answer;

}

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If we assume that all arithmetic operations take O(1) time, then the above function takes O(n) time where n represents the number of bits in the binary representation of the exponent, e.

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Question — What is the expected payoff for Player A?

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Let’s say that instead of 3 rolls of the die, the game allowed only for 1 roll. What is the expected payoff for Player A in this case? It is clearly equal to the expected value of the number that shows up when the die is rolled. And since the die is fair, this expectation is equal to . And Player A has no choice but to accept whatever shows up on the die because there is only one roll possible. Hence, in this game, the expected payoff for Player A is $3.5.

Now, consider what happens if instead of 1 roll, we allow for upto 2 rolls.

Now, if the 1st roll is a 6, obviously Player A would pocket $6 and decide to end the game there itself. Also, if the 1st roll is a 1, Player A can always decide to roll again [because irrespective of what shows up on the 2nd roll, it cannot be less than $1, and hence Player A has nothing to lose by deciding to discard the 1st roll and opting to roll again.]

Now consider what happens if the 1st roll is a 3. Now, Player A is faced with a predicament. What should he do? Should he decide to stop there and go home with $3, or should he discard this 3 in the hope of higher returns in the 2nd and final roll?

This is where having learnt probability well in school helps.

We know that the expected value of the number that shows up when the die is rolled once is 3.5.

Hence, if in the 1st roll, Player A gets a number greater than 3.5 i.e. either 4, 5 or 6, he can decide to stop there and take the corresponding amount of money.

However, if he gets either 1, 2 or 3 in his 1st roll, he should discard that, and decide to roll again, because the expected value of the 2nd roll is 3.5.

Hence, the expected payoff for Player A from a game which allows for at most 2 rolls is: .

(Here, is the probability of getting a 1, 2 or 3 on the 1st roll, in which case the expected payoff becomes 3.5. )

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Now, coming back to our original game, we can decide as to how we should proceed after the die is rolled once.

We know that the expected payoff from 2 rolls of the die is $4.25.

Hence the strategy of Player A should be the following:

- If the 1st roll of the die yields either a 5 or a 6, take the money, and end the game.

- Else, discard the 1st roll, and continue the game.

The expected payoff can be computed easily based on the preceding observations:

Expected payoff for Player A = .

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This is a classic question using the principle of dynamic programming. Let’s say the the game allows for the die to be tossed upto times.

Let denote the expected payoff for Player A in a game that allows upto rolls of the dice.

The strategy for Player A after roll should be the following:

– If the number showing on the dice is greater than , take the money and stop the game.

– Else, discard that roll, and decide to continue on to the next roll.

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The recurrence relation for computing the values can be derived in terms of as was done in the case of the 3-roll example that we saw earlier. Once we have done that, we can calculate the values in the order, , and so on uptil .

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Our task is to select a team of k players from among n players. Now, we have 2 options – Either select player 1 to be part of the team, or leave him out of the team.

If we select player 1 to be part of the team, our task is reduced to selecting players from among players.

And if player 1 is not part of the team, our task is reduced to selecting players from among players.

Hence we have the above result.

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4.

Again our task is to select a team of k players from among n players – .

If is part of the team: We can select such a team in ways.

If is not part of the team, and is part of the team: The problem is reduced to selecting k-1 players from among n-2 players.

If and are not in the team, and is in the team: The problem is reduced to that of selecting k-1 players from among n-3 players.

Proceeding in this manner, we get the above result.

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5.

Again, we have n players at our disposal from among whom we wish to select a team of k players.

Say we select to be the 1st player in our team. (Note that the term “first” actually means nothing since we wish to select a “set” of objects and not an “ordered list”. However the usage of such terminology is useful for the purposes of exposition.)

Now, all we need to do is to select players from among players.

Now, let’s go back a step. What if we select to be the 1st player of the team, and select the remaining players for the team in ways.

We can see that we have n choices for the 1st player of a team.

Hence, we can select a team of k players in .

But again some teams are being repeated.

Given a fixed team, say T, we select that team k times (with each of the k players in the team being selected as the 1st player of the team). Hence we need to divide the previous result by k.

Hence we get the above result.

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We can easily derive some interesting results using the above meaning of .

1.

Our task is to select a team of k players from among n players. We can do this by directly selecting which players are to be part of the team, or by selecting which players are to be left out, giving the above result.

2. =

Look at it this way. We have n players, say , and we want to select a team of k players from among them.

Let us select the players one by one.

For selecting the first player, we have n possible options.

Now having selected one player, we have n-1 possible options for selecting the 2nd player.

Similarly, given that we have selected i players, we have n-i options for selecting the i+1 th player.

Thus, we get that we can select k players from among n players in ways.

But wait!

There are many repetitions in the teams of 11 players that we selected. Consider any particular fixed team of k players. That team is selected by us in ways.

Therefore, we have: .

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Note that given a list of k items, we can have k! permutations of those items. (This can be easily derived recursively.)

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——

We note that whatever handing over of the money takes place between A and B, the total amount of money remains constant at $3.

There are 4 possible states —

State 0 — A has $0

State 1 — A has $1

State 2 — A has $2

State 3 — A has $3.

(Note that the amount of money that B has, in each of the above states, is completely determined by simply stating the amount of money with A since the total amount of money remains constant.)

We get the following transition diagram with probabilities depicted besides the arrows.

The probability that we wish to compute is the probability of reaching State 3 starting initially at State 1.

We can compute that quite easily. Let denote the probability of reaching State 3 from State .

We directly get the following from the above transition probabilities –

.

.

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Therefore, our desired probability, , which means that there is 4/7 probability that the series of tennis matches would end with B becoming bankrupt.

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