以下非常高效的算法使用Heap的方法生成运行时复杂度为O(N!)的N个元素的所有排列:
function permute(permutation) {
var length = permutation.length,
result = [permutation.slice()],
c = new Array(length).fill(0),
i = 1, k, p;
while (i < length) {
if (c[i] < i) {
k = i % 2 && c[i];
p = permutation[i];
permutation[i] = permutation[k];
permutation[k] = p;
++c[i];
i = 1;
result.push(permutation.slice());
} else {
c[i] = 0;
++i;
}
}
return result;
}
console.log(permute([1, 2, 3]));
同样的算法实现为一个空间复杂度为O(N)的生成器:
function* permute(permutation) {
var length = permutation.length,
c = Array(length).fill(0),
i = 1, k, p;
yield permutation.slice();
while (i < length) {
if (c[i] < i) {
k = i % 2 && c[i];
p = permutation[i];
permutation[i] = permutation[k];
permutation[k] = p;
++c[i];
i = 1;
yield permutation.slice();
} else {
c[i] = 0;
++i;
}
}
}
// Memory efficient iteration through permutations:
for (var permutation of permute([1, 2, 3])) console.log(permutation);
// Simple array conversion:
var permutations = [...permute([1, 2, 3])];
性能比较
请随意将您的实现添加到以下benchmark.js测试套件中:
function permute_SiGanteng(input) {
var permArr = [],
usedChars = [];
function permute(input) {
var i, ch;
for (i = 0; i < input.length; i++) {
ch = input.splice(i, 1)[0];
usedChars.push(ch);
if (input.length == 0) {
permArr.push(usedChars.slice());
}
permute(input);
input.splice(i, 0, ch);
usedChars.pop();
}
return permArr
}
return permute(input);
}
function permute_delimited(inputArr) {
var results = [];
function permute(arr, memo) {
var cur, memo = memo || [];
for (var i = 0; i < arr.length; i++) {
cur = arr.splice(i, 1);
if (arr.length === 0) {
results.push(memo.concat(cur));
}
permute(arr.slice(), memo.concat(cur));
arr.splice(i, 0, cur[0]);
}
return results;
}
return permute(inputArr);
}
function permute_monkey(inputArray) {
return inputArray.reduce(function permute(res, item, key, arr) {
return res.concat(arr.length > 1 && arr.slice(0, key).concat(arr.slice(key + 1)).reduce(permute, []).map(function(perm) {
return [item].concat(perm);
}) || item);
}, []);
}
function permute_Oriol(input) {
var permArr = [],
usedChars = [];
return (function main() {
for (var i = 0; i < input.length; i++) {
var ch = input.splice(i, 1)[0];
usedChars.push(ch);
if (input.length == 0) {
permArr.push(usedChars.slice());
}
main();
input.splice(i, 0, ch);
usedChars.pop();
}
return permArr;
})();
}
function permute_MarkusT(input) {
function permutate(array, callback) {
function p(array, index, callback) {
function swap(a, i1, i2) {
var t = a[i1];
a[i1] = a[i2];
a[i2] = t;
}
if (index == array.length - 1) {
callback(array);
return 1;
} else {
var count = p(array, index + 1, callback);
for (var i = index + 1; i < array.length; i++) {
swap(array, i, index);
count += p(array, index + 1, callback);
swap(array, i, index);
}
return count;
}
}
if (!array || array.length == 0) {
return 0;
}
return p(array, 0, callback);
}
var result = [];
permutate(input, function(a) {
result.push(a.slice(0));
});
return result;
}
function permute_le_m(permutation) {
var length = permutation.length,
result = [permutation.slice()],
c = new Array(length).fill(0),
i = 1, k, p;
while (i < length) {
if (c[i] < i) {
k = i % 2 && c[i],
p = permutation[i];
permutation[i] = permutation[k];
permutation[k] = p;
++c[i];
i = 1;
result.push(permutation.slice());
} else {
c[i] = 0;
++i;
}
}
return result;
}
function permute_Urielzen(arr) {
var finalArr = [];
var iterator = function (arrayTaken, tree) {
for (var i = 0; i < tree; i++) {
var temp = arrayTaken.slice();
temp.splice(tree - 1 - i, 0, temp.splice(tree - 1, 1)[0]);
if (tree >= arr.length) {
finalArr.push(temp);
} else { iterator(temp, tree + 1); }
}
}
iterator(arr, 1); return finalArr;
}
function permute_Taylor_Hakes(arr) {
var permutations = [];
if (arr.length === 1) {
return [ arr ];
}
for (var i = 0; i < arr.length; i++) {
var subPerms = permute_Taylor_Hakes(arr.slice(0, i).concat(arr.slice(i + 1)));
for (var j = 0; j < subPerms.length; j++) {
subPerms[j].unshift(arr[i]);
permutations.push(subPerms[j]);
}
}
return permutations;
}
var Combinatorics = (function () {
'use strict';
var version = "0.5.2";
/* combinatory arithmetics */
var P = function(m, n) {
var p = 1;
while (n--) p *= m--;
return p;
};
var C = function(m, n) {
if (n > m) {
return 0;
}
return P(m, n) / P(n, n);
};
var factorial = function(n) {
return P(n, n);
};
var factoradic = function(n, d) {
var f = 1;
if (!d) {
for (d = 1; f < n; f *= ++d);
if (f > n) f /= d--;
} else {
f = factorial(d);
}
var result = [0];
for (; d; f /= d--) {
result[d] = Math.floor(n / f);
n %= f;
}
return result;
};
/* common methods */
var addProperties = function(dst, src) {
Object.keys(src).forEach(function(p) {
Object.defineProperty(dst, p, {
value: src[p],
configurable: p == 'next'
});
});
};
var hideProperty = function(o, p) {
Object.defineProperty(o, p, {
writable: true
});
};
var toArray = function(f) {
var e, result = [];
this.init();
while (e = this.next()) result.push(f ? f(e) : e);
this.init();
return result;
};
var common = {
toArray: toArray,
map: toArray,
forEach: function(f) {
var e;
this.init();
while (e = this.next()) f(e);
this.init();
},
filter: function(f) {
var e, result = [];
this.init();
while (e = this.next()) if (f(e)) result.push(e);
this.init();
return result;
},
lazyMap: function(f) {
this._lazyMap = f;
return this;
},
lazyFilter: function(f) {
Object.defineProperty(this, 'next', {
writable: true
});
if (typeof f !== 'function') {
this.next = this._next;
} else {
if (typeof (this._next) !== 'function') {
this._next = this.next;
}
var _next = this._next.bind(this);
this.next = (function() {
var e;
while (e = _next()) {
if (f(e))
return e;
}
return e;
}).bind(this);
}
Object.defineProperty(this, 'next', {
writable: false
});
return this;
}
};
/* power set */
var power = function(ary, fun) {
var size = 1 << ary.length,
sizeOf = function() {
return size;
},
that = Object.create(ary.slice(), {
length: {
get: sizeOf
}
});
hideProperty(that, 'index');
addProperties(that, {
valueOf: sizeOf,
init: function() {
that.index = 0;
},
nth: function(n) {
if (n >= size) return;
var i = 0,
result = [];
for (; n; n >>>= 1, i++) if (n & 1) result.push(this[i]);
return (typeof (that._lazyMap) === 'function')?that._lazyMap(result):result;
},
next: function() {
return this.nth(this.index++);
}
});
addProperties(that, common);
that.init();
return (typeof (fun) === 'function') ? that.map(fun) : that;
};
/* combination */
var nextIndex = function(n) {
var smallest = n & -n,
ripple = n + smallest,
new_smallest = ripple & -ripple,
ones = ((new_smallest / smallest) >> 1) - 1;
return ripple | ones;
};
var combination = function(ary, nelem, fun) {
if (!nelem) nelem = ary.length;
if (nelem < 1) throw new RangeError;
if (nelem > ary.length) throw new RangeError;
var first = (1 << nelem) - 1,
size = C(ary.length, nelem),
maxIndex = 1 << ary.length,
sizeOf = function() {
return size;
},
that = Object.create(ary.slice(), {
length: {
get: sizeOf
}
});
hideProperty(that, 'index');
addProperties(that, {
valueOf: sizeOf,
init: function() {
this.index = first;
},
next: function() {
if (this.index >= maxIndex) return;
var i = 0,
n = this.index,
result = [];
for (; n; n >>>= 1, i++) {
if (n & 1) result[result.length] = this[i];
}
this.index = nextIndex(this.index);
return (typeof (that._lazyMap) === 'function')?that._lazyMap(result):result;
}
});
addProperties(that, common);
that.init();
return (typeof (fun) === 'function') ? that.map(fun) : that;
};
/* permutation */
var _permutation = function(ary) {
var that = ary.slice(),
size = factorial(that.length);
that.index = 0;
that.next = function() {
if (this.index >= size) return;
var copy = this.slice(),
digits = factoradic(this.index, this.length),
result = [],
i = this.length - 1;
for (; i >= 0; --i) result.push(copy.splice(digits[i], 1)[0]);
this.index++;
return (typeof (that._lazyMap) === 'function')?that._lazyMap(result):result;
};
return that;
};
// which is really a permutation of combination
var permutation = function(ary, nelem, fun) {
if (!nelem) nelem = ary.length;
if (nelem < 1) throw new RangeError;
if (nelem > ary.length) throw new RangeError;
var size = P(ary.length, nelem),
sizeOf = function() {
return size;
},
that = Object.create(ary.slice(), {
length: {
get: sizeOf
}
});
hideProperty(that, 'cmb');
hideProperty(that, 'per');
addProperties(that, {
valueOf: function() {
return size;
},
init: function() {
this.cmb = combination(ary, nelem);
this.per = _permutation(this.cmb.next());
},
next: function() {
var result = this.per.next();
if (!result) {
var cmb = this.cmb.next();
if (!cmb) return;
this.per = _permutation(cmb);
return this.next();
}
return (typeof (that._lazyMap) === 'function')?that._lazyMap(result):result;
}
});
addProperties(that, common);
that.init();
return (typeof (fun) === 'function') ? that.map(fun) : that;
};
/* export */
var Combinatorics = Object.create(null);
addProperties(Combinatorics, {
C: C,
P: P,
factorial: factorial,
factoradic: factoradic,
permutation: permutation,
});
return Combinatorics;
})();
function permute_Technicalbloke(inputArray) {
if (inputArray.length === 1) return inputArray;
return inputArray.reduce( function(accumulator,_,index){
permute_Technicalbloke([...inputArray.slice(0,index),...inputArray.slice(index+1)])
.map(value=>accumulator.push([inputArray[index],value]));
return accumulator;
},[]);
}
var suite = new Benchmark.Suite;
var input = [0, 1, 2, 3, 4];
suite.add('permute_SiGanteng', function() {
permute_SiGanteng(input);
})
.add('permute_delimited', function() {
permute_delimited(input);
})
.add('permute_monkey', function() {
permute_monkey(input);
})
.add('permute_Oriol', function() {
permute_Oriol(input);
})
.add('permute_MarkusT', function() {
permute_MarkusT(input);
})
.add('permute_le_m', function() {
permute_le_m(input);
})
.add('permute_Urielzen', function() {
permute_Urielzen(input);
})
.add('permute_Taylor_Hakes', function() {
permute_Taylor_Hakes(input);
})
.add('permute_Combinatorics', function() {
Combinatorics.permutation(input).toArray();
})
.add('permute_Technicalbloke', function() {
permute_Technicalbloke(input);
})
.on('cycle', function(event) {
console.log(String(event.target));
})
.on('complete', function() {
console.log('Fastest is ' + this.filter('fastest').map('name'));
})
.run({async: true});
<script src="https://cdnjs.cloudflare.com/ajax/libs/lodash.js/4.17.4/lodash.min.js"></script>
<script src="https://cdnjs.cloudflare.com/ajax/libs/platform/1.3.4/platform.min.js"></script>
<script src="https://cdnjs.cloudflare.com/ajax/libs/benchmark/2.1.4/benchmark.min.js"></script>
Chrome 48的运行时结果:
99.152ms此算法
153.115 MarkusT女士
401.255 js-combinatorics女士
497.037 Urielzen女士
925.669 Oriol女士
1026.571 SiGanteng女士
2529.841毫秒分隔
3992.622毫秒的猴子
4514.665 Oriol女士
