add permutations backtracking

Author: amejiarosario

book/chapters/backtracking.adoc

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

book/chapters/sample.adoc

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