浙江大学DS05_Ch10a_A高级数据结构课件

The Multiprocessor Case – N jobs on P processors 〖Example〗 job 〗 time P=3
An Optimal Solution j1 j2 j3
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j1 3
j2 5
j3 6
j4 10
j5 11
j6 14
j7 15
j8 18
j9 20
is the size of the character set */
Discussion 9: In what case that there are not likely to be any savings even using Huffman codes?
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§1 Greedy Algorithms
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Research Project 4
Huffman Codes (30)
As a professor who gives the final exam problem on Huffman codes, I am encountering a big problem: the Huffman codes are NOT unique. The students are submitting all kinds of codes, and I need a computer program to help me determine which ones are correct and which ones are not. Detailed requirements can be downloaded from /ds/
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§1 Greedy Algorithms
1. Huffman Codes – for file compression 〖Example〗 Suppose our text is a string of length 1000 that 〗
comprises the characters a, u, x, and z. Then it will take 8000 ? bits to store the string as 1000 one-byte characters. We may encode the symbols as a = 00, u = 01, x = 10, z = 11. For example, aaaxuaxz is encoded as 0000001001001011. Then the space Notice that we have only taken by the string distinct characters in be 2000 bits + space for 4 with length 1000 will that string. code table. /* log C bits are we need in a standard encoding where C Hence needed only 2 bits to identify them. frequency ::= number of occurrences of a symbol. In string aaaxuaxz , f(a) = 4, f(u) = 1, f(x) = 2, f(z) = 1. The size of the coded string can be reduced using variable-length 00010110010111 codes, for example, a = 0, u = 110, x = 10, z = 111.
Representation of the original code in a binary tree /* trie */ 0 0 a 1 u 1 0 x 1 z
If character Ci is at depth di and occurs fi times, then the cost of the code = Σ di fi . Cost ( aaaxuaxz 0000001001001011 ) = 2×4 + 2×1 + 2×2 + 2×1 = 16 × × × ×
CHAPTER 10
ALGORITHM DESIGN TECHNIQUES
§1 Greedy Algorithms
Optimization Problems: Given a set of constrains and an optimization function. Solutions that satisfy the constrains are called feasible solutions. A feasible solution for which the optimization function has the best possible value is called an optimal solution. The Greedy Method: Make the best decision at each stage, under some greedy criterion. A decision made in one stage is not changed in a later stage, so each decision should assure feasibility.
In general:
0
ji1
ti1
N
ji2
ti1 + ti2
ji3
N
……
ti1 + ti2 + ti3
N
Total Cost C = ∑ ( N − k + 1)t ik = ( N + 1)∑ t ik − ∑ k ⋅ t ik
k =1 k =1 k =1
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§1 Greedy Algorithms
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§1 Greedy Algorithms
2. A Simple Scheduling Problem The Single Processor Case Given N jobs j1 , j2 , …, jN , their running times t1 , t2 , …, tN , and one processor. Schedule jobs to minimize the average completion time. /* assume nonpreemptive scheduling */
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0 te s
1
4 18
t4 34 10来自840 nl
18
1 sa t
t a sp 10 nl
s
11011 : 1100 : 01 : 11010
nl
1
4
0 t
4
1
4
nl
1
0 nl
1
1 s
3
3 Cost = 3×10 + 2×15 × × + 2×12 + 5×3 × × s 3 + 4×4 + 2×13 × × + 5×1 × = 146
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§1 Greedy Algorithms
Note: Greedy algorithm works only if the local optimum is equal to the global optimum. Greedy algorithm does not guarantee optimal solutions. However, it generally produces solutions that are very close in value (heuristics) to the optimal, and hence is intuitively appealing when finding the optimal solution takes too much time.
Discussion 10: What must the tree look like if we are to decode unambiguously?
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§1 Greedy Algorithms
Huffman’s Algorithm (1952)
void Huffman ( PriorityQueue heap[ ], int C ) { consider the C characters as C single node binary trees, and initialize them into a min heap; for ( i = 1; i < C; i++ ) { create a new node; /* be greedy here */ delete root from min heap and attach it to left_child of node; delete root from min heap and attach it to right_child of node; weight of node = sum of weights of its children; /* weight of a tree = sum of the frequencies of its leaves */ insert node into min heap; } }