Design & Analysis of Algorithm Dynamic Programming Informatics Department Parahyangan Catholic University

Introduction We have seen some algorithm design principles, such as divide-and-conquer, brute force, and greedy Brute force is widely applicable, but inefficient Divide-and-conquer and greedy are fast, but only applicable on very specific problems. Dynamic Programming is somewhere in between them, while still providing polynomial time complexity, it is widely applicable.

Dynamic Programming (DP) Similar to divide-and-conquer, DP solves problem by combining the solutions of its sub-problems. The term “programming” here refers to a tabular method DP solves a sub-problem, then saves its solution in a table

Finding n-th fibonacci number 0 n=0 F(n) = 1 n=1 F(n-1) + F(n-2) n>1 FIBONACCI (n) if (n==0) return 0 else if (n==1) return 1 else return FIBONACCI(n-1) + FIBONACCI(n-2) Recursive solution :

Finding n-th fibonacci number Recursive solution : F(5) F(4) F(3) F(3) F(2) F(1) F(2) F(1) F(2) F(0) F(1) F(0) F(1) F(0) F(1)

Finding n-th fibonacci number Memoization : Maintain a table to store sub-problem’s solution // Initially : Arr[0] = 0 // Arr[1] = 1 // Arr[2..n] = -1 FIBONACCI (n) if (Arr[n] != -1) return Arr[n] else Arr[n]=FIBONACCI(n-1) + FIBONACCI(n-2) solution with memoization:

Finding n-th fibonacci number solution with memoization: F(5) F(4) F(3) =2 F(3) F(2) =1 F(1) F(2) F(0) F(1) n 1 2 3 4 5 F(n) -1 1 2 3 5

Finding n-th fibonacci number Bottom-up solution : Use the natural ordering of sub-problems, solve them one-by-one starting from the “smallest” one FIBONACCI (n) Arr[0] = 0 Arr[1] = 1 if(n>1) for i=2 to n do Arr[i] = Arr[i-1] + Arr[i-2] return Arr[n] Bottom-up solution:

Time complexity Recursive solution : FIBONACCI (n) if (n==0) return 0 else if (n==1) return 1 else return FIBONACCI(n-1) + FIBONACCI(n-2) Every instance has ≤ 2 recursive calls Height = n F(n-2) F(n-3) F(n-4) F(n) F(n-1) Therefore, time complexity is O(2n)

Time complexity Bottom-up solution: FIBONACCI (n) Arr[0] = 0 if(n>1) for i=2 to n do Arr[i] = Arr[i-1] + Arr[i-2] return Arr[n] Bottom-up solution: There is a loop that iterates ≤ n times, each doing a constant amount of work. So the time complexity is O(n)

Rod Cutting Problem Example : n=4 Serling Enterprises buys long steel rods and cuts them into shorter rods, which it then sells. Each cut is free, however different rod length sells for different price. The management of Sterling Enterprises wants to know the best way to cut up the rods. We assume that we know : length 1 2 3 4 5 6 7 8 9 10 price 17 20 24 30 Example : n=4 (n-1) possible cut locations  2n-1 ways of cutting

Rod Cutting Problem Example : n=4 length 1 2 3 4 5 6 7 8 9 10 price 17 20 24 30 Example : n=4 1 5 =7 9 =9 1 8 =9 1 5 =7 BEST 5 =10 1 5 =7 1 8 =9 1 =4

Rod Cutting Problem 1 i n Consider a rod of length n, and we cut a rod of length i Then we left with a rod of length n-i Naturally, we want to optimize the selling price of the remaining rod

Rod Cutting Problem Recursive solution : ROD-CUTTING (n) if (n==0) return 0 else best = -∞ for i=1 to n do best = MAX(best, price[i]+ROD-CUTTING(n-i)) return best

Same problem as recursive Fibonacci Rod Cutting Problem Recursive solution : RC(4) RC(2) RC(1) RC(0) RC(3) RC(2) RC(1) RC(0) RC(1) RC(0) RC(0) RC(1) RC(0) RC(0) RC(0) Same problem as recursive Fibonacci RC(0)

Rod Cutting Problem Exercise Solution with memoization: // Initially : Arr[0] = 0 // Arr[1..n] = -∞ ROD-CUTTING (n) if Arr[n] ≥ 0 return Arr[n] else best = -∞ for i=1 to n do best = MAX(best, price[i]+ROD-CUTTING(n-i)) Arr[n] = best return best Exercise Write a bottom-up solution for Rod Cutting problem !

What is the time complexity for the bottom-up solution ? Recursive solution : ROD-CUTTING (n) if (n==0) return 0 else best = -∞ for i=1 to n do best = MAX(best, price[i]+ROD-CUTTING(n-i)) return best Every instance has ≤ n recursive calls, and the depth of the recursive tree is O(n). So the time complexity is O(nn) What is the time complexity for the bottom-up solution ?

Shortest Path in DAG DAG = Directed Acyclic Graph Example : S A C B D 2 1 4 6 3

Shortest Path in DAG S A C B D E 2 1 4 6 3 Sort using topological sort

Shortest Path in DAG Starting from S, suppose we want to reach node D The only way to reach D is either through B or C dist(D) = MIN(dist(B)+1 , dist(C)+3) 3 2 4 6 1 1 S C A B D E 1 2

Shortest Path in DAG A similar relation can be written for every node. As we have seen before, it is best to use bottom-up way of computing dist, that is from “left most” node to “right most” node DAG-SHORTEST-PATH () initialize all dist[.] to ∞ dist[S] = 0 for each vertex v except S in left-to-right order do for each edge (u,v) do dist[v] = MIN(dist[v] , dist[u] + weight(u,v))

Dynamic Programming DAG’s shortest path is a very general technique. We model many other problems into DAG problem. Example #1: Fibonacci Example #2 : Rod Cutting 2 1 3 4 5 2 1 3 4

Properties of Dynamic Programming Problem can be divided into smaller sub-problems Optimal substructure: an optimal solution to the problem contains within its optimal solutions to sub-problems Overlapping sub-problems: the space of sub-problems is “small”, in the sense that a recursive algorithm for the problem solves the same sub-problems over and over, rather than always generating new sub-problems

Dynamic Programming V.S. Divide-and-Conquer And so on…

Longest Increasing Subsequence Given a sequence of numbers a1, a2, a3, … , an. A subsequence is any subset of these numbers taken in order of the form ai1, ai2, ai3…, aik, where 1 ≤ i1 < i2 < … < ik ≤ n, and an increasing subsequence is one which the numbers are getting strictly larger. The task is to find the increasing subsequence of greatest length. Example: 5, 2, 8, 6, 3, 6, 9, 7

Longest Increasing Subsequence How do we model this problem into a DAG ? any number ai can precede aj iff ai < aj Consider this number, LIS that ends here must be either: subsequence consist of “6” alone LIS that ends at “5” + “6” LIS that ends at “2” + “6” LIS that ends at “”3” + “6” 5 2 8 6 3 6 9 7

Longest Increasing Subsequence How do we find the LIS of sequence 5, 2, 8, 6, 3, 6, 9, 7 ? Does the optimal solution always ends at “7” ? // Initially L[1..n] = 1 LONGEST-INCREASING-SUBSEQUENCE () for j=2 to n do for each i<j such that ai < aj do if(L[j] < 1 + L[i]) then L[j] = 1 + L[i] return maximum of L[1..n] This algorithm only gives the sequence’s length. How do we find the actual sequence ?

Reconstructing a solution We can extend the dynamic programming approach to record not only the optimal value for each sub-problem, but also the choice that led to that value // Initially L[1..n] = 1 // Initially prev[1..n] = 0 LONGEST-INCREASING-SUBSEQUENCE () for j=2 to n do for each i<j such that ai < aj do if(L[j] < 1 + L[i]) then L[j] = 1 + L[i] prev[j] = i return maximum of L[1..n] and array prev

Exercise : Yuckdonald’s Yuckdonald’s is considering opening a series of restaurants along Quaint Valley Highway (QVH). The n possible locations are along a straight line, and the distance of these locations from the start of QVH are, in miles and in increasing order, m1, m2, …, mn. The constraints are as follows: At each location, Yuckdonald’s may open at most one restaurant. The expected profit from opening a restaurant at location i is pi > 0 Any two restaurants should be at least k miles apart, where k is a positive integer