bubbleSort.js

function bubbleSort(arr, cmp) {
  var len = arr.length, tmp, n;
  if (typeof cmp !== 'function')
    cmp = function(a, b) {
      return a > b;
    };
  do {
    n = 0;
    for (var i = 1; i < len; i++) {
      if (cmp(arr[i], arr[i-1])) continue;
      n = i;
      tmp = arr[i-1];
      arr[i-1] = arr[i];
      arr[i] = tmp;
    }
    len = n;
  } while (len > 0);
  return arr;
}

hdfSort.js

function hdfSort(arr, comparer) {
  var len = arr.length;
  if (len < 2) return arr;
  var tmp = arr;
  if (typeof comparer !== 'function')
    comparer = function(a, b) {
      return a > b;
    };
  var l2 = Math.ceil(len / 2);
  var ret = new Array(len);
  var smallest, greatest;
  for (var i = 0; i < l2; i++) {
    for (var i2 = 0; i2 < len; i2++) {
      if (tmp[i2] == null) continue;
      if (smallest == null || comparer(tmp[smallest], tmp[i2])) smallest = i2;
      if (greatest == null || comparer(tmp[i2], tmp[greatest])) greatest = i2;
    }
    ret[i] = tmp[smallest];
    ret[len-i-1] = tmp[greatest];
    tmp[smallest] = null;
    tmp[greatest] = null;
    greatest = smallest = null;
  }
  return ret;
}

insertSort.js

function insertSort(arr, cmp) {
  var len = arr.length, tmp, j;
  if (typeof cmp !== 'function')
    cmp = function(a, b) {
      return a > b;
    };
  for (var i = 1; i < len; i++) {
    tmp = arr[i];
    j = i - 1;
    while (j >= 0 && cmp(arr[j], tmp)) {
      arr[j+1] = arr[j];
      j--;
    }
    arr[j+1] = tmp;
  }
  return arr;
}

mergeSort.js

var mergeSort = (function () {
  var cmp = {};
  var default_cmp = function(a, b) {
    return a < b;
  };

  function merger(arr) {
    if (arr.length < 2) return arr;
    let pivot = Math.ceil(arr.length / 2);
    return merge(merger(arr.slice(0, pivot)), merger(arr.slice(pivot)));
  }

  function merge(left, right) {
    var res = [], l = 0, r = 0;
    while(l < left.length && r < right.length)
      res.push(cmp(left[l], right[r]) ? left[l++] : right[r++]);
    return res.concat(left.slice(l), right.slice(r));
  }

  return function (arr, comparer) {
    cmp = (typeof comparer === 'function') ? comparer : default_cmp;
    return merger(arr);
  };
}());

quickSort.js

Array.prototype.quickSort = function(cmp) {
  var len = this.length;
  if (len < 2) return this;
  var arr = this;
  if (typeof cmp !== 'function')
    cmp = function(a, b) {
      return a < b;
    };
  var less = [], greater = [];
  var pivot = arr.splice(Math.floor(len-- / 2), 1);
  for (var i = 0; i < len; i++) {
    if (cmp(arr[i], pivot))
      less.push(arr[i]);
    else
      greater.push(arr[i]);
  }
  return less.quickSort(cmp).concat(pivot, greater.quickSort(cmp));
};

Benchmark:
http://jsben.ch/qVHSf
Interestingly, depending on how large the array is, the results can be totally different. The takeaway however is, that the larger the array, the more recursion costs, and at some point it even offsets the cost of extra iterations. (Probably due to a lack of Tail Call Optimization.)