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Commit e8cd69d

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add permutations backtracking
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‎book/chapters/backtracking.adoc‎

Lines changed: 67 additions & 3 deletions
Original file line numberDiff line numberDiff line change
@@ -24,14 +24,78 @@ It stops evaluating a path as soon as some of the conditions are broken and move
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However, it an only be applied if a quick test can be run to tell if a candidate will contribute to a valid solution.
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****
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== How to develop Backtracking algorithms?
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Let's do an example to explain better how backtracking works.
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// https://leetcode.com/problems/combination-sum/description/
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Backtracking algorithms can be tricky to get right or reason about but we are going to follow this recipe to make it easier.
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.Steps to create backtracking algorithms
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. Iterate through all the elements in the input
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. Make a change
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. Call recursive function
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. Test if the current change is a solution
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. Revert back the change (backtracking)
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Let's do an exercise to explain better how backtracking works.
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// https://leetcode.com/problems/combination-sum/description/
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== Permutations
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> Return all the permutations (without repetitions) of a given word.
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For instace, if are given the word `art` we can have the following permutions
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----
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[ [ 'art' ],
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[ 'atr' ],
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[ 'rat' ],
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[ 'rta' ],
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[ 'tra' ],
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[ 'tar' ] ]
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----
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We already solved this problem using an <<Getting all permutations of a word, iterative program>>, now let's do it using backtracking.
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[graphviz, Recursive Fibonacci call tree with dp, svg]
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....
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digraph g {
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node [shape = record,height=.1];
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art[label = "<f0> A|<f1> R|<f2> T"];
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art1[label = "<f0> A|<f1> R|<f2> T"];
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art2[label = "<f0> A|<f1> R|<f2> T", color="red"];
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atr[label = "<f0> A|<f1> T|<f2> R", color="red"];
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rat[label = "<f0> R|<f1> A|<f2> T"];
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rat1[label = "<f0> R|<f1> A|<f2> T", color="red"];
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rta[label = "<f0> R|<f1> T|<f2> A", color="red"];
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tra[label = "<f0> T|<f1> R|<f2> A"];
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tra1[label = "<f0> T|<f1> R|<f2> A", color="red"];
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tar[label = "<f0> T|<f1> A|<f2> R", color="red"];
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art:f0 -> art1:f0 [ label = "1. swap A/A"];
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art1:f0 -> art2:f0 [ label = "2. swap R/R"];
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art2:f2 -> art1:f1 [ label = "3.", color="grey", fontcolor="grey"];
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art1:f2 -> atr:f0 [ label = "4. swap R/T"];
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atr:f2 -> art1:f2 [ label = "5.", color="grey", fontcolor="grey"];
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art1:f1 -> art:f0 [ label = "6.", color="grey", fontcolor="grey"];
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art:f1 -> rat:f0 [ label = "7. swap A/R"];
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rat:f0 -> rat1:f0 [ label = "8. swap A/A"];
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rat1:f2 -> rat:f1 [ label = "9.", color="grey", fontcolor="grey"];
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rat:f2 -> rta:f0 [ label = "10. swap A/T"];
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rta:f2 -> rat:f2 [ label = "11.", color="grey", fontcolor="grey"];
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rat:f2 -> art:f2 [ label = "12.", color="grey", fontcolor="grey"];
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art:f2 -> tra:f0 [ label = "13. swap A/T"];
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tra:f0 -> tra1:f0 [ label = "14. swap R/R"];
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tra1:f2 -> tra:f2 [ label = "15.", color="grey", fontcolor="grey"];
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tra:f2 -> tar:f0 [ label = "16. swap R/A"];
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tar:f2 -> tra:f2 [ label = "17.", color="grey", fontcolor="grey"];
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tra:f2 -> art:f2 [ label = "18.", color="grey", fontcolor="grey"];
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}
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....
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‎book/chapters/sample.adoc‎

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@@ -256,6 +256,41 @@ graph G {
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}
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....
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[graphviz, Recursive Fibonacci call tree with dp, svg]
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....
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digraph g {
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node [shape = record,height=.1];
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art[label = "<f0> A|<f1> R|<f2> T"];
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art1[label = "<f0> A|<f1> R|<f2> T"];
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art2[label = "<f0> A|<f1> R|<f2> T", color="red"];
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atr[label = "<f0> A|<f1> T|<f2> R", color="red"];
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rat[label = "<f0> R|<f1> A|<f2> T"];
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rat1[label = "<f0> R|<f1> A|<f2> T", color="red"];
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rta[label = "<f0> R|<f1> T|<f2> A", color="red"];
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tra[label = "<f0> T|<f1> R|<f2> A"];
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tra1[label = "<f0> T|<f1> R|<f2> A", color="red"];
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tar[label = "<f0> T|<f1> A|<f2> R", color="red"];
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"art":f0 -> "art1":f0 [ label = "1. swap A/A"];
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"art1":f0 -> "art2":f0 [ label = "2. swap R/R"];
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"art2":f2 -> "art1":f1 [ label = "3", color="grey"];
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"art1":f2 -> "atr":f0 [ label = "4. swap R/T"];
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"atr":f2 -> "art1":f2 [ label = "5", color="grey"];
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"art1":f1 -> "art":f0 [ label = "6", color="grey"];
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"art":f1 -> "rat":f0 [ label = "7. swap A/R"];
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"rat":f0 -> "rat1":f0 [ label = "8. swap A/A"];
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"rat1":f2 -> "rat":f1 [ label = "9", color="grey"];
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"rat":f2 -> "rta":f0 [ label = "10. swap A/T"];
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"art":f2 -> "tra":f0 [ label = "swap A/T"];
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"tra":f0 -> "tra1":f0 [ label = "swap R/R"];
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"tra":f2 -> "tar":f0 [ label = "swap R/A"];
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}
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....
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// ------------------

‎lab/exercises/medium/permute.js‎

Lines changed: 9 additions & 6 deletions
Original file line numberDiff line numberDiff line change
@@ -13,16 +13,16 @@ var permute = function (nums) {
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return result;
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};
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function backtrack(nums, result, first = 0) {
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function backtrack(nums, result, index = 0) {
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// console.log(Array(first).fill(" ").join(""), JSON.stringify({nums, first}));
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if (first === nums.length - 1) {
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if (index === nums.length - 1) {
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// console.log(nums);
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result.push(nums.slice());
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} else {
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for (let i = first;i < nums.length; i++) {
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swap(nums, first,i);
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backtrack(nums, result, first + 1);
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swap(nums, first,i);
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for (let current = index;current < nums.length; current++) {
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swap(nums, index,current);
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backtrack(nums, result, index + 1);
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swap(nums, index,current);
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}
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}
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};
@@ -51,5 +51,8 @@ function test() {
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[3, 2, 1],
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[3, 1, 2]
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]);
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console.log(permute(Array.from('art')));
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}
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test();

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