PERMUTATION GENERATION METHODS Robert ... - cs.Princeton
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PERMUTATION GENERATION METHODS Robert ... - cs.Princeton
Q: Does the order of the edges in the input matter? A: Of course! Q: How? A: It depends on ... Compute all perms of a gl
Now is the time For all
good PERMUTATION men GENERATION METHODS
To come to the aid Of their party
Robert Sedgewick Princeton University
Motivation PROBLEM Generate all N! permutations of N elements Q: Why? Basic research on a fundamental problem
Compute exact answers for insights into combinatorial problems Structural basis for backtracking algorithms Numerous published algorithms, dating back to 1650s CAVEATS N is between 10 and 20 can be the basis for extremely dumb algorithms
processing a perm often costs much more than generating it
N
million/sec billion/sec trillion/sec
N is between 10 and 20 number of perms
14 87178291200
13 6227020800
12 479001600
11 39916800
weeks
day
hours
minutes
seconds
hours
minutes
minute
seconds
seconds
10 3628800
15 1307674368000
months
minutes
insignificant
16 20922789888000
days
hours
years
months
17 355687428096000 18 6402373705728000
days
month
years impossible
19 121645100408832000 20 2432902008176640000
Digression: analysis of graph algorithms Typical graph-processing scenario: input graph as a sequence of edges (vertex pairs) build adjacency-lists representation run graph-processing algorithm Q: Does the order of the edges in the input matter? A: Of course! Q: How? A: It depends on the graph Q: How?
2 There are 2V graphs, so full employment for algorithm analysts
€
Digression (continued)
Ex: compute a spanning forest (DFS, stop when V vertices hit)
best case cost: V (right edge appears first on all lists) Complete digraph on V vertices worst case: V2
€
average: V ln V (Kapidakis, 1990) Same graph with single outlier
€ worst case: O(V2 ) average: O(V2 )
Can we estimate the average for€a given graph? € the edges to speed things up? Is there a simple way to reorder
What impact does edge order have on other graph algorithms?
Digression: analysis of graph algorithms Insight needed, so generate perms to study graphs
No shortage of interesting graphs with fewer than 10 edges
Algorithm to compute average generate perms, run graph algorithm Goal of analysis faster algorithm to compute average
C B D A
C D B A
D C B A
B D C A
D B C A
Method 1: backtracking
D C A B
C D A B
C A D B
A C D B
D A C B
A D C B
B D A C
D B A C
D A B C
A D B C
B A D C
A B D C
B D A D
C B A D
C A B D
A C B D
B A C D
Compute all perms of a global array by exchanging each element to the end, then recursively permuting the others exch (int i, int j) { int t = p[i]; p[i] = p[j]; p[j] = t; } generate(int N) { int c; if (N == 1) doit(); for (c = 1; c
