[ACE] Dynamic Programming

Overview

There are various general methods for finding solutions to problems.
Common ones include:
  • Brute force - “generate and test”
  • Divide and conquer
  • Heuristics
  • Dynamic Programming

Brute Force

This is roughly “generate and test”, generate all potential solutions and test for which ones are actual solutions.
This method is extremely inefficient, as is O(n!), but can be useful in some small cases.

Divide and Conquer

Recursively, break the problem into smaller pieces, solve them, and put them back together.
Merge-sort and Quick-sort were classic examples of this.

Heuristics

Two common types:
  • Decisions within a procedure that gives optimal answer to make it go faster.
  • Decisions within a procedure that might not give optimal answers, but are designed to give good answers that are impractical to obtain otherwise.
    Examples:
  • “Admissible heuristic” in A* search - decreases the search time compared to plain search
  • “pick a random pivot” in quick sort

Greedy algorithms

  • A common “heuristic” is to be greedy
  • Take the decision that looks best in short term
  • Prim’s algorithm for constructing a Minimal Spanning Tree is a greedy algorithm: it just adds the shortest edge without worrying about the overall structure, without looking ahead. It makes a locally optimal choice at each step.
  • Sometimes greedy algorithms can still give optimal answers
  • Sometimes greedy algorithms cannot guarantee to give optimal answers, example is “Change-Giving”
  • enter image description here

“Min-Coins-Change-Giving”

“Obviously” this can be solved by enumerating all possible subsets of S, selecting those that sum to K, and picking a subset with the fewest elements.
  • But with n coins there are 2^n subsets and so this “generate-and-test” naive algorithm is exponential in the worst case.
  • Can we do better? Firstly, consider greedy strategies:
  • Greedy strategy: start with the largest coin which is available; for the remaining change, again pick the largest coin; and so on.
  • Coins = {50, 50, 20 ,20, 20, 2, 2, 2 ,2, 2} Change = 60
  • Greedy method picks {50, 2, 2, 2, 2, 2} - 6coins
  • But can do it with {20, 20, 20} - 3 coins
    That’s why we need DP!

Dynamic Programming(DP)

DP is a general method that can be suitable when the optimal solutions satisfy a “decomposition property”:
  • Splitting an optimal solution into sub-solutions
  • So optimal solutions can be built out of optimal solutions of (smaller) sub-problems
  • So solve small sub-problems first and build up towards the full solution

Subset-Sum

Given multi-set S of positive integers x[i] and a target K:
Is there a subset of S that sums to exactly k?
The simple underlying idea is to suppose we have found all the best answers for the coins x[0],…x[i-1] and then want to also add the effect of one more coin x[i]:
  • if some subset summed to m, then with the inclusion x[i] we can also find a subset that sums to m+x[i]
  • and with one more coin than was recorded as possible for m
  • If a set of coins had already been found then take the one that gives the minimum.

Complexity

  • Outer loop has to consider all the coins –O(n)
  • Inner loop scans the entire array Y – hence O(k)
  • Overall is O(n K), much better than O(2^n)

Is this all? No!

In some ACM competition and OJ like LeetCode, there are a lot of question related to DP.

Coins change 2


My answer:





House robber



This is also a easy DP problem.
Think that:
  • On every house the robber robbed, the benefit is either: the benefit of this house + the benefit of the second previous house or the benefit of previous house(if the robber doesn’t rob this house)
  • There is a DP equation: DP[i] = max(DP[i-1], DP[i-2] + nums[i]);

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