Dynamic programming is discussed in next post i.e.
1) Overlapping Subproblems
Memotization
Tabulation
2) Optimal Substructure
What is subsequences ?
A subsequence is a sequence that appears in the same relative order, but not necessarily contiguous.
The list of all subsequences for the word "apple" would be "e, l, le, p, pe, pl, ple, p, pe, pl, ple, pp, ppe, ppl, pple, a, ae, al, ale, ap, ape, apl, aple, ap, ape, apl, aple, app, appe, appl, apple".
- Longest Increasing Subsequence
Input : arr[] = {3, 10, 2, 1, 20} Output : Length of LIS = 3 The longest increasing subsequence is 3, 10, 20 Input : arr[] = {3, 2} Output : Length of LIS = 1 The longest increasing subsequences are {3} and {2} Input : arr[] = {50, 3, 10, 7, 40, 80} Output : Length of LIS = 4 The longest increasing subsequence is {3, 7, 40, 80}https://www.youtube.com/watch?v=1RpMc3fv0y4
LCS Problem Statement: Given two sequences, find the length of longest subsequence present in both of them. For example, “abc”, “abg”, “bdf”, “aeg”, ‘”acefg”, .. etc are subsequences of “abcdefg”. So a string of length n has 2^n different possible subsequences.
It is a classic computer science problem, the basis of diff (a file comparison program that outputs the differences between two files), and has applications in bioinformatics.
Examples:
LCS for input Sequences “ABCDGH” and “AEDFHR” is “ADH” of length 3.
LCS for input Sequences “AGGTAB” and “GXTXAYB” is “GTAB” of length 4.
LCS for input Sequences “ABCDGH” and “AEDFHR” is “ADH” of length 3.
LCS for input Sequences “AGGTAB” and “GXTXAYB” is “GTAB” of length 4.
https://www.youtube.com/watch?v=43P0xZp3FU4&list=PLeIMaH7i8JDjMEB-b2I8NGcKMFZc85djW&index=10
Given two strings str1 and str2 and below operations that can performed on str1. Find minimum number of edits (operations) required to convert ‘str1’ into ‘str2’.
- Insert
- Remove
- Replace
All of the above operations are of equal cost.
Examples:
Examples:
Input: str1 = "geek", str2 = "gesek" Output: 1 We can convert str1 into str2 by inserting a 's'. Input: str1 = "cat", str2 = "cut" Output: 1 We can convert str1 into str2 by replacing 'a' with 'u'. Input: str1 = "sunday", str2 = "saturday" Output: 3 Last three and first characters are same. We basically need to convert "un" to "atur". This can be done using below three operations. Replace 'n' with 'r', insert t, insert a
https://www.youtube.com/watch?v=b6AGUjqIPsA
Partition a set into two subsets such that the difference of subset sums is minimum
Given a set of integers, the task is to divide it into two sets S1 and S2 such that the absolute difference between their sums is minimum.
If there is a set S with n elements, then if we assume Subset1 has m elements, Subset2 must have n-m elements and the value of abs(sum(Subset1) – sum(Subset2)) should be minimum.
Example:
Input: arr[] = {1, 6, 11, 5}
Output: 1
Explanation:
Subset1 = {1, 5, 6}, sum of Subset1 = 12
Subset2 = {11}, sum of Subset2 = 11
Count number of ways to cover a distance
Given a distance ‘dist, count total number of ways to cover the distance with 1, 2 and 3 steps.
Examples:
Input: n = 3 Output: 4 Below are the four ways 1 step + 1 step + 1 step 1 step + 2 step 2 step + 1 step 3 step Input: n = 4 Output: 7
Find the longest path in a matrix with given constraints
Given a n*n matrix where all numbers are distinct, find the maximum length path (starting from any cell) such that all cells along the path are in increasing order with a difference of 1.
We can move in 4 directions from a given cell (i, j), i.e., we can move to (i+1, j) or (i, j+1) or (i-1, j) or (i, j-1) with the condition that the adjacent cells have a difference of 1.
Example:
Input: mat[][] = {{1, 2, 9}
{5, 3, 8}
{4, 6, 7}}
Output: 4
The longest path is 6-7-8-9.
Dynamic Programming | Set 25 (Subset Sum Problem)
Given a set of non-negative integers, and a value sum, determine if there is a subset of the given set with sum equal to given sum.
Examples: set[] = {3, 34, 4, 12, 5, 2}, sum = 9
Output: True //There is a subset (4, 5) with sum 9.
Given weights and values of n items, put these items in a knapsack of capacity W to get the maximum total value in the knapsack. In other words, given two integer arrays val[0..n-1] and wt[0..n-1] which represent values and weights associated with n items respectively. Also given an integer W which represents knapsack capacity, find out the maximum value subset of val[] such that sum of the weights of this subset is smaller than or equal to W. You cannot break an item, either pick the complete item, or don’t pick it (0-1 property).
Dynamic Programming | Set 37 (Boolean Parenthesization Problem)
Given a boolean expression with following symbols.
Symbols
'T' ---> true
'F' ---> false
And following operators filled between symbols
Operators
& ---> boolean AND
| ---> boolean OR
^ ---> boolean XOR
Count the number of ways we can parenthesize the expression so that the value of expression evaluates to true.
Let the input be in form of two arrays one contains the symbols (T and F) in order and other contains operators (&, | and ^}
Examples:
Input: symbol[] = {T, F, T}
operator[] = {^, &}
Output: 2
The given expression is "T ^ F & T", it evaluates true
in two ways "((T ^ F) & T)" and "(T ^ (F & T))"
Input: symbol[] = {T, F, F}
operator[] = {^, |}
Output: 2
The given expression is "T ^ F | F", it evaluates true
in two ways "( (T ^ F) | F )" and "( T ^ (F | F) )".
Input: symbol[] = {T, T, F, T}
operator[] = {|, &, ^}
Output: 4
The given expression is "T | T & F ^ T", it evaluates true
in 4 ways ((T|T)&(F^T)), (T|(T&(F^T))), (((T|T)&F)^T)
and (T|((T&F)^T)).
Shortest Common Supersequence
Given two strings str1 and str2, find the shortest string that has both str1 and str2 as subsequences.
Examples:
Input: str1 = "geek", str2 = "eke" Output: "geeke" Input: str1 = "AGGTAB", str2 = "GXTXAYB" Output: "AGXGTXAYB"
Dynamic Programming | Set 18 (Partition problem)
Partition problem is to determine whether a given set can be partitioned into two subsets such that the sum of elements in both subsets is same.
Examples
arr[] = {1, 5, 11, 5}
Output: true
The array can be partitioned as {1, 5, 5} and {11}
arr[] = {1, 5, 3}
Output: false
The array cannot be partitioned into equal sum sets.
Dynamic Programming | Set 13 (Cutting a Rod)
Given a rod of length n inches and an array of prices that contains prices of all pieces of size smaller than n. Determine the maximum value obtainable by cutting up the rod and selling the pieces. For example, if length of the rod is 8 and the values of different pieces are given as following, then the maximum obtainable value is 22 (by cutting in two pieces of lengths 2 and 6)
length | 1 2 3 4 5 6 7 8 -------------------------------------------- price | 1 5 8 9 10 17 17 20
And if the prices are as following, then the maximum obtainable value is 24 (by cutting in eight pieces of length 1)
length | 1 2 3 4 5 6 7 8 -------------------------------------------- price | 3 5 8 9 10 17 17 20
Given a value N, if we want to make change for N cents, and we have infinite supply of each of S = { S1, S2, .. , Sm} valued coins, how many ways can we make the change? The order of coins doesn’t matter.
For example, for N = 4 and S = {1,2,3}, there are four solutions: {1,1,1,1},{1,1,2},{2,2},{1,3}. So output should be 4. For N = 10 and S = {2, 5, 3, 6}, there are five solutions: {2,2,2,2,2}, {2,2,3,3}, {2,2,6}, {2,3,5} and {5,5}. So the output should be 5.
Given an input string and a dictionary of words, find out if the input string can be segmented into a space-separated sequence of dictionary words. See following examples for more details.
This is a famous Google interview question, also being asked by many other companies now a days.
This is a famous Google interview question, also being asked by many other companies now a days.
Consider the following dictionary
{ i, like, sam, sung, samsung, mobile, ice,
cream, icecream, man, go, mango}
Input: ilike
Output: Yes
The string can be segmented as "i like".
Input: ilikesamsung
Output: Yes
The string can be segmented as "i like samsung"
or "i like sam sung".
(Maximum Product Cutting)
Given a rope of length n meters, cut the rope in different parts of integer lengths in a way that maximizes product of lengths of all parts. You must make at least one cut. Assume that the length of rope is more than 2 meters.
Examples:
Input: n = 2 Output: 1 (Maximum obtainable product is 1*1) Input: n = 3 Output: 2 (Maximum obtainable product is 1*2) Input: n = 4 Output: 4 (Maximum obtainable product is 2*2) Input: n = 5 Output: 6 (Maximum obtainable product is 2*3) Input: n = 10 Output: 36 (Maximum obtainable product is 3*3*4)
(Dice Throw)
Given n dice each with m faces, numbered from 1 to m, find the number of ways to get sum X. X is the summation of values on each face when all the dice are thrown.
The Naive approach is to find all the possible combinations of values from n dice and keep on counting the results that sum to X.
This problem can be efficiently solved using Dynamic Programming (DP).
Let the function to find X from n dice is: Sum(m, n, X)
The function can be represented as:
Sum(m, n, X) = Finding Sum (X - 1) from (n - 1) dice plus 1 from nth dice
+ Finding Sum (X - 2) from (n - 1) dice plus 2 from nth dice
+ Finding Sum (X - 3) from (n - 1) dice plus 3 from nth dice
...................................................
...................................................
...................................................
+ Finding Sum (X - m) from (n - 1) dice plus m from nth dice
So we can recursively write Sum(m, n, x) as following
Sum(m, n, X) = Sum(m, n - 1, X - 1) +
Sum(m, n - 1, X - 2) +
.................... +
Sum(m, n - 1, X - m)
(Box Stacking Problem)
You are given a set of n types of rectangular 3-D boxes, where the i^th box has height h(i), width w(i) and depth d(i) (all real numbers). You want to create a stack of boxes which is as tall as possible, but you can only stack a box on top of another box if the dimensions of the 2-D base of the lower box are each strictly larger than those of the 2-D base of the higher box. Of course, you can rotate a box so that any side functions as its base. It is also allowable to use multiple instances of the same type of box.
Source: http://people.csail.mit.edu/bdean/6.046/dp/. The link also has video for explanation of solution.
(Egg Dropping Puzzle)
The following is a description of the instance of this famous puzzle involving n=2 eggs and a building with k=36 floors.
Suppose that we wish to know which stories in a 36-story building are safe to drop eggs from, and which will cause the eggs to break on landing. We make a few assumptions:
…..An egg that survives a fall can be used again.
…..A broken egg must be discarded.
…..The effect of a fall is the same for all eggs.
…..If an egg breaks when dropped, then it would break if dropped from a higher floor.
…..If an egg survives a fall then it would survive a shorter fall.
…..It is not ruled out that the first-floor windows break eggs, nor is it ruled out that the 36th-floor do not cause an egg to break.
…..A broken egg must be discarded.
…..The effect of a fall is the same for all eggs.
…..If an egg breaks when dropped, then it would break if dropped from a higher floor.
…..If an egg survives a fall then it would survive a shorter fall.
…..It is not ruled out that the first-floor windows break eggs, nor is it ruled out that the 36th-floor do not cause an egg to break.
If only one egg is available and we wish to be sure of obtaining the right result, the experiment can be carried out in only one way. Drop the egg from the first-floor window; if it survives, drop it from the second floor window. Continue upward until it breaks. In the worst case, this method may require 36 droppings. Suppose 2 eggs are available. What is the least number of egg-droppings that is guaranteed to work in all cases?
The problem is not actually to find the critical floor, but merely to decide floors from which eggs should be dropped so that total number of trials are minimized.
The problem is not actually to find the critical floor, but merely to decide floors from which eggs should be dropped so that total number of trials are minimized.
C++ program for weighted job scheduling using Dynamic Programming. #include
|
No comments:
Post a Comment