拉斯维加斯算法&蒙特卡罗算法_杨劲松

• the probability of Majority method for return ture: p>1/2 • the probability of Majority2 method for return ture: p+(1-p)p = 1-(1-p)² > 3/4
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Las Vegas Does N-Queens ： � Solution4 Solution4：
• random permutation vectors
�a
random permutation vectors
3 7 2 8 1 4 6
two vector flags for diagonal attacks
�
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Las Vegas Does N-Queens ： � Solution5 Solution5：
• GridSearch
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
� Retain information about cell
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failtimes =2
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Las Vegas Does N-Queens
� The efficiency with different step
solution2
solution3
• step: the number of randomly placed queen • p: success probability • s: the average number of nodes search access for a successful
� � � � do {LV(x,y,stopVegas); Backtracking(n-stopVegas) { ...
if(fail) { failtimes++; trace-back(failtimes) }
� } � }while(LV(x,y)is success)
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failtimes =1
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Las Vegas Does N-Queens ： � Solution3 Solution3：
• backtrack n steps according failtimes
� � � � do {LV(x,y,stopVegas); Backtracking(n-stopVegas) { ...
（1，1，1……，1，1），（1，1，1……，1，2） ，（1，1，1……，2，2），……，（n，n，n…… ，n，n）。
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N-Queens ： � The solution space tree tree：
root a[1][1] a[2][1] 2 3 3 1 2 2 2 a[1][2] 1 2 2 2 a[1][3] 1 2 2
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Las Vegas Does N-Queens ： � Solution2 Solution2：
• combine the trace-back and Las Vegas algorithm � do � { //Placeing the part of
queens with random raw
• backtrack n steps according failtime
� � � � do {LV(x,y,stopVegas); Backtracking(n-stopVegas) { ...
if(fail) { failtimes++; trace-back(failtimes) }
� } � }while(LV(x,y)is success)
1 1 1 1 1 1 1
1 1
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Monte Carlo Algorithm � Characteristic
• running time is deterministic. • output may be incorrect with a certain (typically small) probability.
if(fail) { failtimes++; trace-back(failtimes) }
� } � }while(LV(x,y)is success)
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Las Vegas Does N-Queens ： � Solution3 Solution3：
• backtrack n steps according failtimes
3
......
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Las Vegas Does N-Queens ： Poor efficiency � Solution1 Solution1：
� do { � LV(x,y){ � try to place the queen in random
column
�} � }while(LV(x,y)is success)
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华南师范大学计算机学院 – 计算机算法
拉斯维加斯算法 &蒙特卡罗算法
作者：杨劲松
2013-6-21
Outline
� Probabilistic Algorithm � Las Vegas Algorithm � Probabilistic Algorithm
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Probabilistic Algorithm
INPUT
ALGORITHM
OUTPUT
RANDOM NUMBERS
� Algorithm takes a source of random numbers and makes random choices during execution; � Behavior can vary even on a fixed input;
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Las Vegas Does N-Queens ： � Solution3 Solution3：
• backtrack n steps according failtimes
� � � � do {LV(x,y,stopVegas); Backtracking(n-stopVegas) { ...
� public static boolean majority(int[]t, int n) � { rnd = new Random(); � int i=rnd.random(n)+1; � int x=t[i]; int k=0; � for (int j=1;j<=n;j++) if (t[j]==x) k++; � return (k>n/2); � � }
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Monte Carlo Algorithm � The main element of array
• In an array whose size is N, the main element is a occurs more than N / 2 times . • eg: [ 1,2,3,1,3,1,1] , the main element is 1.
百度文库
Monte Carlo Algorithm
table1.Monte Carlo versus Las Vegas
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REFERENCES
[1]Rolfe T J. Las Vegas does n-queens[J]. ACM SIGCSE Bulletin, 2006, 38(2): 37-38. [2]何月梅 ,刘翠霞 ,张青 . 拉斯维加斯方法解 n 后问题的研究 [J]. 邯郸学院 学报 ,2008,03:35-37.
if(fail) { failtimes++; trace-back(failtimes) }
� } � }while(LV(x,y)is success)
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Las Vegas Does N-Queens ： � Solution3 Solution3：
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Probabilistic Algorithm
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Las Vegas Algorithm
� Randomized algorithm � Correct results � void obstinate(Object x, Object y) �{ � bool success= false; � while (!success) success=LV(x,y); �}
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Monte Carlo Algorithm � The main element of array
• Repeated calls to improve the correct probability
� public static boolean Majority2(int []t, int n) � {// call Majority method 2 times � if (Majority(t,n)) return true; � else return majority(t,n); � }
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N-Queens � Placing n queens on an n×n chessboard.
� No two queens attack each other.
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N-Queens � Solution vector ： X[i] represents the positions of the i-th queen in the x[i]-th column ，so，the solution vector is（X[1],X[2],X[3],……， X[n]） ：（ nn） � Solution space space：（ ：（n
� LV(x,y,stopVegas); � Backtracking(nstopVegas) � }while(LV(x,y)is success)
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Las Vegas Does N-Queens ： � Solution2 Solution2：
• combine the trace-back and Las Vegas algorithm
the efficiency with different step
• step: the number of randomly placed queen • p: success probability the average number of nodes search access for • s: a successful