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Cakewalk

# Problem:

Given N sticks of lengths L[1], L[2], ... L[N] and a positive integer D. Two sticks can be paired if the difference of their lengths is at most D. Pair up as many sticks as possible such that each stick is used at most once.

# Quick Explanation:

Sort the sticks by their lengths and let L be the sorted array.

If L[1] and L[2] cannot be paired, then the stick L[1] is useless. Otherwise, there exists an optimal pairing where L[1] gets paired with L[2].

This gives rise to the following algorithm:

 numpairs = 0 for ( i = 1; i < N; ) if (L[i] >= L[i+1] -D) // L[1] and L[2] can be paired numpairs++, // pair them up i += 2; else i++; // eliminate L[1] 

# Justifications:

• If L[1] and L[2] cannot be paired then
L[1] < L[2] - D
But, L[2] <= L[i] for every i > 1
So    L[1] < L[i] - D for every i > 1
Hence, L[1] cannot be paired with anyone.

• If L[1] and L[2] can be paired.
Consider any optimal pairing and it can be transformed to a pairing where L[1] and L[2] are paired.
a) If the optimal pairing pairs L[1] with L[2] then we are done.
b) If only one of L[1] and L[2] is paired with someone, then we can replace that pair by (L[1], L[2]).
c) If both L[1] and L[2] are paired and L[1] is paired with L[n] and L[2] with L[m], then we might as well form pairs (L[1], L[2]) and (L[n], L[m]).
This is because
L[2] <= L[m] <= L[2] + D
L[2] <= L[n] <= L[1] + D <= L[2] + D
⇒   -D <= L[m] - L[n] <= D

# Setter's Solution:

Can be found here

# Tester's Solution:

Can be found here

This question is marked "community wiki".

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accept rate: 0%

1★garakchy
1.1k163048

(22 Jul '13, 00:06)

If L[1] and L[2] cannot be paired then L[1] < L[2] + D If L[1]=L[2]+D-1. Then (L[2]-L[1])=1-D <=D It should be L[1]<L[2]-D .

(22 Jul '13, 00:26) 2★

@maggu you are right. fixed.

(22 Jul '13, 00:33)

 0 Can anyone point my mistake ...? http://www.codechef.com/viewsolution/4329369 Thanks. :) answered 14 Jul '14, 15:55 3★tech_boy 1.2k●4●19●31 accept rate: 7%
 0 why my solution is giving TLE? http://www.codechef.com/viewsolution/4361912 answered 21 Jul '14, 13:47 1★arjun045 1 accept rate: 0%
 0 //what is wrong with this #include #include #define ll long long using namespace std; int main() { ll n,d; cin>>n>>d; ll a[n]; for(ll i=0;i>a[i]; sort(a,a+n); ll cnt=0; for(ll i=0;i
 0 @anichavan20 you are accessing nth element of array, which is throwing some garbage value.. when i=n-1,you try to compute a[n]-a[n-1]... solve this.. you may try this http://www.codechef.com/viewsolution/4361912 answered 27 Jun '15, 21:20 25●1●1●4 accept rate: 0%
 0 https://www.codechef.com/viewsolution/12375437 Can someone please help what is wrong with this?? Thank you. answered 31 Dec '16, 18:21 3★rs_14 1 accept rate: 0%
 0 I have done with this can you please provide some more test case for this problem. answered 13 Jan '17, 10:07 1 accept rate: 0%
 0 Why my code is wrong : https://www.codechef.com/viewsolution/14081239 answered 05 Jun '17, 16:42 3 accept rate: 0%
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question asked: 22 Jul '13, 00:01

question was seen: 5,751 times

last updated: 05 Jun '17, 16:42