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 2There 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 wi and value

vi. 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 vi : wi

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 si and finishing at fi.

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, [ai, bi]

Choosing the minimum number of points {pj} 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 ConquerDivide

Break a problem into sub problem(s)

ConquerSolve all sub problem(s)

CombineSolve the problem using the results of the sub

problem(s)

Example 1: Merge Sort, Quick Sort

1 ~ N

X+1 ~ N1 ~ X

1 ~ Y Y+1 ~ X X+1 ~ Z Z+1 ~ N

1 2 3 N

Example 2: Tower of HanoiFind 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 = XP mod M

O(P) is easy: XP = X * XP-1

0 P 2≦ ≦ 31 -1 ……

X2N = XN * XN

X2N+1 = X * XN * XN

Example 4: L-pieces

A 2N * 2N 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 QueryGiven N numbers, find the maximum value from ai to

bi.

O(NQ) ?

If we know RMQ(3,7), can we find RMQ(3,10) faster?

RMQ(ai, bi) = MAX( RMQ(ai, ci), RMQ(ci+1, bi) )

Segment Tree

Conclusion

Try to divide the original problem to some easier sub-problems.

You will see some similar ideas in DP.