Gary Sham HKOI 2010 Greedy, Divide and Conquer

Greedy Algorithm Solve the problem by the “BEST” choice. To find the global optimal through local optimal choices, and solve the sub-problem.

Example 1: Coins There are 7 kinds of coins: $0.1, $0.2, $0.5, $1, $2, $5, $10 What is the minimum number needed to pay $18 ? Use the greatest possible coin every time.

Coins 2 There are 7 kinds of coins: $0.1, $0.2, $0.5, $2, $5, $9, $10 What is the minimum number needed to pay $18 ? Greedy? $10 + $5 + $2 + $2 Actually…… $9 + $9

Example 2: Fractional Knapsack There are N objects, each with weight w i and value v i. Any amount ( including fractions ) of each item can be taken provided that the total weight does not exceed W. How much of each item should we take in order to maximize the total value? Consider v i : w i

0-1 Knapsack problem Similar to Fractional Knapsack, but you can only choose to pick the whole item or not. Greedy?

Example 3: Activity There are n activities, starting at time s i and finishing at f i. Choose the maximum number of activities so that they do not overlap each other. Greedy?

Example 4: Diamond Chain To find the maximum interval sum. When will you choose a negative value?

Example 5: Advertisement There are n intervals, [a i, b i ] Choosing the minimum number of points {p j } so that each interval contains at least one of the points

Example 6: Egyptian fraction An Egyptian fraction is a sum of distinct unit fraction Given a fraction and express it as an Egyptian fraction

Conclusion Hard to prove or disprove. Usually easy to code. Usually efficient. Try to guess.

Divide and Conquer Divide Break a problem into sub problem(s) Conquer Solve all sub problem(s) Combine Solve the problem using the results of the sub problem(s)

Example 1: Merge Sort, Quick Sort 1 ~ N X+1 ~ N1 ~ X 1 ~ YY+1 ~ XX+1 ~ ZZ+1 ~ N 123 N

Example 2: Tower of Hanoi Find the minimum number of steps to move N stacks from Pag0 to Pag2. How to move N stacks? Move N-1 stacks from Pag0 to Pag1. Move the Nth stack to Pag2. Move N-1 stacks from Pag1 to Pag2. How to move N-1 stacks? ……

Example 3: Big Mod R = X P mod M O(P) is easy: X P = X * X P-1 0 ≦ P ≦ …… X 2N = X N * X N X 2N+1 = X * X N * X N

Example 4: L-pieces A 2 N * 2 N square board with 1 hole. How to place L-pieces to cover it? What is the sub-problem?

Solution 2N-1 * 2N-1 board with 1 hole

Example 5: Range Maximum Query Given N numbers, find the maximum value from a i to b i. O(NQ) ? If we know RMQ(3,7), can we find RMQ(3,10) faster? RMQ(a i, b i ) = MAX( RMQ(a i, c i ), RMQ(c i +1, b i ) ) Segment Tree

Conclusion Try to divide the original problem to some easier sub-problems. You will see some similar ideas in DP.