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Fix: Updated notes

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riyabansal98 2019-10-10 17:17:06 +05:30
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Recursion and Backtracking/1.md → Recursion and Backtracking/Recursion_new.md
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@ -1,9 +1,10 @@

Recursion Recursion
---------- ----------
Recursion - process of function calling itself Recursion - process of function calling itself
directly or indirectly. directly or indirectly.
_Steps involved:_ __Steps involved:__
- Base case - Base case
- Self Work - Self Work
- Recursive Calls - Recursive Calls
@ -34,9 +35,9 @@ def pow(n, k):
if k == 0: return 1 if k == 0: return 1
return n*pow(n, k - 1) return n*pow(n, k - 1)
``` ```
_Time Complexity_: $O(n)$ __Time Complexity__: $O(n)$
_Optimised solution:_ __Optimised solution:__
```python ```python
def pow(n, k): def pow(n, k):
if k == 0: return 1 if k == 0: return 1
@ -53,7 +54,7 @@ To allow reuse of answers.
<img src="https://user-images.githubusercontent.com/35702912/66316190-d30e1f00-e934-11e9-8089-85c6dc69baa7.jpg" data-canonical-src="https://user-images.githubusercontent.com/35702912/66316190-d30e1f00-e934-11e9-8089-85c6dc69baa7.jpg" width="400" /> <img src="https://user-images.githubusercontent.com/35702912/66316190-d30e1f00-e934-11e9-8089-85c6dc69baa7.jpg" data-canonical-src="https://user-images.githubusercontent.com/35702912/66316190-d30e1f00-e934-11e9-8089-85c6dc69baa7.jpg" width="400" />
_Time Complexity_ (assuming all multiplications are O(1))? $O(\log_2 k)$ __Time Complexity__ (assuming all multiplications are O(1))? $O(\log_2 k)$
Break it into 3 parts? k//3 and take care of mod1 and mod2. Break it into 3 parts? k//3 and take care of mod1 and mod2.
@ -89,11 +90,10 @@ def subsets(A, i, aux):
<img src="https://user-images.githubusercontent.com/35702912/66323471-7a914e80-e941-11e9-84a9-11a333ac4f77.jpg" width="400" <img src="https://user-images.githubusercontent.com/35702912/66323471-7a914e80-e941-11e9-84a9-11a333ac4f77.jpg" width="400"
/> />
How many leaf nodes? $2^n$ - one for each subset How many leaf nodes? $2^n$ - one for each subset
How many total nodes? $2^{n+1} - 1$ How many total nodes? $2^{n+1} - 1$
Complexity? $O(2^n)$ __Time Complexity__: $O(2^n)$
Subsets using Iteration Subsets using Iteration
----------------------- -----------------------
@ -149,6 +149,12 @@ For Array [0,1,2,3,4] Subsets in Lexicographical order,
But don't print when going left - because already printed in parent. But don't print when going left - because already printed in parent.
<img src="https://user-images.githubusercontent.com/35702912/66468106-3d44d200-eaa3-11e9-96e7-c6a050be1219.jpg" width="400"
/>
<img src="https://user-images.githubusercontent.com/35702912/66468119-42098600-eaa3-11e9-8f24-237be2a91d12.jpg" width="400"
/>
```python ```python
def subsets(A, i, aux, p): def subsets(A, i, aux, p):
if p: print(aux) if p: print(aux)
@ -158,8 +164,8 @@ def subsets(A, i, aux, p):
no_take = subsets(A, i+1, aux, False) no_take = subsets(A, i+1, aux, False)
``` ```
time: $O(2^n)$ __Time Complexity__: $O(2^n)$
Space: $O(n^2)$, because we're creating new aux arrays. __Space Complexity__: $O(n^2)$, because we're creating new aux arrays.
-- -- -- --
Number of Subsets with a given Sum Number of Subsets with a given Sum
@ -173,6 +179,9 @@ The subsetSum problem can be divided into two subproblems.
- Include the current element in the sum and recur (i = i + 1) for the rest of the array - Include the current element in the sum and recur (i = i + 1) for the rest of the array
- Exclude the current element from the sum and recur (i = i + 1) for the rest of the array. - Exclude the current element from the sum and recur (i = i + 1) for the rest of the array.
<img src="https://user-images.githubusercontent.com/35702912/66469192-11c2e700-eaa5-11e9-9094-252ce842464a.jpg" width="400"
/>
```python ```python
def subsetSum(A,N,cur_sum, i, target): def subsetSum(A,N,cur_sum, i, target):
if i == N: if i == N:
@ -191,6 +200,8 @@ For eg,
target = 6 target = 6
1, 2, 3 is good, but 1, 2, 3 is good, but
1, 2, 3, -1, 1 is also good. 1, 2, 3, -1, 1 is also good.
__Time Complexity__: $O(2^n)$
__Space Complexity__: $O(n)$
Number of Subsets with a given Sum (Repetition Allowed) Number of Subsets with a given Sum (Repetition Allowed)
--------------- ---------------
@ -200,7 +211,7 @@ Number of Subsets with a given Sum (Repetition Allowed)
The subsetSum2 problem can be divided into two subproblems. The subsetSum2 problem can be divided into two subproblems.
- Include the current element in the sum and recur for the rest of the array. Here the value of i is not incremented to incorporate the condition of including multiple occurances of a element. - Include the current element in the sum and recur for the rest of the array. Here the value of i is not incremented to incorporate the condition of including multiple occurances of a element.
- Exclude the current element from the sum and recur (i = i + 1) for the rest of the array. - Exclude the current element from the sum and recur (i = i + 1) for the rest of the array.
![IMG_0040](https://user-images.githubusercontent.com/35702912/66470200-c27db600-eaa6-11e9-8744-ca572d6000e1.jpg)
```python ```python
def subsetSum2(A,N,cur_sum, i, target): def subsetSum2(A,N,cur_sum, i, target):
if i == N: if i == N:
@ -214,6 +225,8 @@ def subsetSum2(A,N,cur_sum, i, target):
no_take = subsetSum2(A,N,cur_sum, i+1, target) no_take = subsetSum2(A,N,cur_sum, i+1, target)
return take + no_take return take + no_take
``` ```
__Time Complexity__ : $O(2 ** (Target/MinElement))$
__Space Complexity__: $O(Target/Min Element)$