# Delete some bits and count

Consider all 2^n different binary strings of length n and assume n > 2. You are allowed to delete exactly b < n/2 bits from each of the binary strings, leaving strings of length n-b remaining. The number of distinct strings remaining depends on which bits you delete. Assuming your aim is to leave as few remaining different strings as possible, this challenge is to write code to compute how few can you leave as a function of n.

Example, n=3 and b = 1. You can leave only the two strings 11 and 00.

For n=9 and b = 1,2,3,4 we have 70,18,6,2

For n=8 and b = 1,2,3 we have 40,10,4

For n=7 and b = 1,2,3 we have 20,6,2

For n=6 and b = 1,2 we have 12,4

For n=5 and b = 1,2 we have 6,2

This question was originally posed by me in 2014 in a different form on MO.

Input and output

Your code should take in an integern and output a single integer for each value of b starting at b = 0 and increasing.

Score

Your score is the largest n for which your code completes for all b < n/2 in under a minute on my Linux based PC. In case of tie breaks, the largest b your code gets to for the joint largest n wins. In case of tie breaks on that criterion too, the fastest code for the largest values of n and b decides. If the times are within a second or two of each other, the first posted answer wins.

Languages and libraries

You can use any language of library you like. Because I have to run your code, it would help if it was free (as in beer) and worked in Linux.

• I'm assuming b > 0 as additional input-requirement? Or would n=3 and b=0 simply output 2^n as result? – Kevin Cruijssen May 28 '18 at 9:12
• @KevinCruijssen It should output 2^n indeed. – user9207 May 28 '18 at 9:13
• Also, you say the input is a single n and a single b, but the score is the largest n for which the code completes all b < n/2 in under a minute. Wouldn't it be better to have a single input n in that case, and output all results for 0 <= b < n/2? Or should we provide two programs/functions: one taking two inputs n and b, and one taking only input n and outputting all results in the range 0 <= b < n/2? – Kevin Cruijssen May 28 '18 at 9:18
• Well, I had already upvoted your challenge, so can't do it again. :) Although I have no idea how to calculate this efficiently (efficient O algorithms were something I've always been bad at.. and one of the few subjects at IT college I had to redo a couple of times), it does seem like a very interesting challenge. I'm curious to see what answers people come up with. – Kevin Cruijssen May 28 '18 at 9:47
• Is there a working example? It would be a good place to start, both in terms of correctness, but also for comparison of speed. – maxb May 28 '18 at 10:39

Python 2.7 / Gurobi n=9

This solution is very straight usage of Gurobi's ILP solver for the boolean Mixed-Integer Problems (MIP).

The only trick is to take out symmetry in 1's complement to halve the problem sizes.

Using Gurobi LLC's limited time "free" licence we are restricted to 2000 constraints, but solving 10 del 1 is well outside the 60 second time-limit anyway on my laptop.

from gurobipy import *
from itertools import combinations

def mincover(n,d):
bs = pow(2,n-1-d)
m = Model()
m.Params.outputFlag = 0
b = {}
for i in range(bs):
b[i] = m.addVar(vtype=GRB.BINARY, name="b%d" % i)
m.update()
for row in range(pow(2,n-1)):
x = {}
for i in combinations(range(n), n-d):
v = 0
for j in range(n-d):
if row & pow(2,i[j]):
v += pow(2,j)
if v >= bs:
v = 2*bs-1-v
x[v] = 1
m.addConstr(quicksum(b[i] for i in x.keys()) >= 1)
m.setObjective(quicksum(b[i] for i in range(bs) ), GRB.MINIMIZE)
m.optimize()
return int(round(2*m.objVal,0))

for n in range(4,10):
for d in range((n//2)+1):
print n, d, mincover(n,d)

UPDATE+CORR: 10,2 has optimal solution size 31 (see e.g.) Gurobi shows no symmetric solution of size 30 exists (returns problem infeasible) .. [my attempt to show asymmetric feasibility at 30 remained inconclusive after 9.5hrs runtime] e.g. bit patterns of integers 0 7 13 14 25 28 35 36 49 56 63 64 95 106 118 128 147 159 170 182 195 196 200 207 225 231 240 243 249 252 255 or 0 7 13 14 19 25 28 35 36 49 56 63 64 95 106 118 128 159 170 182 195 196 200 207 225 231 240 243 249 252 255

• You broke the "fastest claimed infinite bounty" record? – user202729 Jun 10 '18 at 1:15
• I don't see any bounty here, what do you mean? – jayprich Jun 10 '18 at 7:16
• @user202729 Yes.. I set it too low. I should have set it at n = 10 :) – user9207 Jun 10 '18 at 8:20
• Actually solving it at n=9 is not an easy thing. That's why OP use an existing library (which is supposed to be better than a hand-written solution, like mine). – user202729 Jun 10 '18 at 10:42
• Thanks @ChristianSievers I see MO claim that 10,2 has only asymmetric optima which I cannot refute nor verify. If I remove the symmetry assumption shortcut which works up to n=9 it turns out Gurobi can still solve up to n=9 in the time required. – jayprich Jun 13 '18 at 17:23

# C++, n=6

Brute force with some small optimizations.

#include<cassert>
#include<iostream>
#include<vector>

// ===========
/** Helper struct to print binary representation.
std::cout<<bin(str,len) prints (str:len) == the bitstring
represented by last (len) bits of (str).
*/
struct bin{
int str,len;
bin(int str,int len):str(str),len(len){}
};
std::ostream& operator<<(std::ostream& str,bin a){
if(a.len)
return str<<bin(a.str>>1,a.len-1)<<char('0'+(a.str&1));
else if(a.str)
return str<<"...";
else
return str;
}
// ===========

/// A patten of (len) bits of ones.
int constexpr pat1(int len){
return (1<<len)-1;
}

// TODO benchmark: make (res) global variable?

/**Append all distinct (subseqs+(sfx:sfxlen)) of (str:len)
with length (sublen) to (res).
*/
void subseqs_(
int str,int len,int sublen,
int sfx,int sfxlen,
std::vector<int>& res
){
// std::cout<<"subseqs_ : str = "<<bin(str,len)<<", "
// "sublen = "<<sublen<<", sfx = "<<bin(sfx,sfxlen)<<'\n';

assert(len>=0);

if(sublen==0){ // todo remove some branches can improve perf?
res.push_back(sfx);
return;
}else if(sublen==len){
res.push_back(str<<sfxlen|sfx);
return;
}else if(sublen>len){
return;
}

if(str==0){
res.push_back(sfx);
return;
}

int nTrail0=0;
for(int ncut;str&&nTrail0<sublen;

++nTrail0,
ncut=__builtin_ctz(~str)+1, // cut away a bit'0' of str
// plus some '1' bits
str>>=ncut,
len-=ncut
){
ncut=__builtin_ctz(str)+1; // cut away a bit'1' of str
subseqs_(str>>ncut,len-ncut,sublen-nTrail0-1,
sfx|1<<(sfxlen+nTrail0),sfxlen+nTrail0+1,
res
); // (sublen+sfxlen) is const. TODO global var?
}

if(nTrail0+len>=sublen) // this cannot happen if len<0
res.push_back(sfx);
}

std::vector<int> subseqs(int str,int len,int sublen){
assert(sublen<=len);
std::vector<int> res;
if(__builtin_popcount(str)*2>len){ // too many '1's, flip [todo benchmark]
subseqs_(pat1(len)^str,len,sublen,0,0,res);
int const p1sublen=pat1(sublen);
for(int& r:res)r^=p1sublen;
}else{
subseqs_(str,len,sublen,0,0,res);
}
return res;
}

// ==========

/** Append all distinct (supersequences+(sfx:sfxlen)) of (str:len)
with length (suplen) to (res).
Define (a) to be a "supersequence" of (b) iff (b) is a subsequence of (a).
*/
void supseqs_(
int str,int len,int suplen,
int sfx,int sfxlen,
std::vector<int>& res
){
assert(suplen>=len);

if(suplen==0){
res.push_back(sfx);
return;
}else if(suplen==len){
res.push_back(str<<sfxlen|sfx);
return;
}

int nTrail0; // of (str)
if(str==0){
res.push_back(sfx);
// it's possible that the supersequence is '0000..00'
nTrail0=len;
}else{
// str != 0 -> str contains a '1' bit ->
// supersequence cannot be '0000..00'
nTrail0=__builtin_ctz(str);
}
// todo try nTrail0=__builtin_ctz(str|1<<len), eliminates a branch
// and conditional statement

for(int nsupTrail0=0;nsupTrail0<nTrail0;++nsupTrail0){
// (nsupTrail0+1) last bits of supersequence matches with
// nsupTrail0 last bits of str.
supseqs_(str>>nsupTrail0,len-nsupTrail0,suplen-1-nsupTrail0,
sfx|1<<(nsupTrail0+sfxlen),sfxlen+nsupTrail0+1,
res);
}

int const strMatch=str?nTrail0+1:len;
// either '1000..00' or (in case str is '0000..00') the whole (str)

for(int nsupTrail0=suplen+strMatch-len;nsupTrail0-->nTrail0;){
// because (len-strMatch)<=(suplen-1-nsupTrail0),
// (nsupTrail0<suplen+strMatch-len).

// (nsupTrail0+1) last bits of supersequence matches with
// (strMatch) last bits of str.
supseqs_(str>>strMatch,len-strMatch,suplen-1-nsupTrail0,
sfx|1<<(nsupTrail0+sfxlen),sfxlen+nsupTrail0+1,
res);
}

// todo try pulling constants out of loops
}

// ==========

int n,b;
std::vector<char> done;
unsigned min_undone=0;

int result;
void backtrack(int nchoice){
assert(!done[min_undone]);
++nchoice;
std::vector<int> supers_s;
for(int s:subseqs(min_undone,n,n-b)){
// obviously (s) is not chosen. Try choosing (s)
supers_s.clear();
supseqs_(s,n-b,n,0,0,supers_s);
for(unsigned i=0;i<supers_s.size();){
int& x=supers_s[i];
if(!done[x]){
done[x]=true;
++i;
}else{
x=supers_s.back();
supers_s.pop_back();
}
}

unsigned old_min_undone=min_undone;
while(true){
if(min_undone==done.size()){
// found !!!!
result=std::min(result,nchoice);
goto label1;
}
if(not done[min_undone])
break;
++min_undone;
}
if(nchoice==result){
// backtrack more will only give worse result
goto label1;
}

// note that nchoice is already incremented
backtrack(nchoice);

label1: // undoes the effect of (above)
for(int x:supers_s)
done[x]=false;
min_undone=old_min_undone;
}
}

int main(){
std::cin>>n>>b;

done.resize(1<<n,0);
result=1<<(n-b); // the actual result must be less than that

backtrack(0);
std::cout<<result<<'\n';
}

Run locally:

[user202729@archlinux golf]$g++ -std=c++17 -O2 delbits.cpp -o delbits [user202729@archlinux golf]$ time for i in $(seq 1 3); do ./delbits <<< "6$i"; done
12
4
2

real    0m0.567s
user    0m0.562s
sys     0m0.003s
[user202729@archlinux golf]$time ./delbits <<< '7 1' ^C real 4m7.928s user 4m7.388s sys 0m0.173s [user202729@archlinux golf]$ time for i in $(seq 2 3); do ./delbits <<< "7$i"; done
6
2

real    0m0.040s
user    0m0.031s
sys     0m0.009s
• Mostly to encourage others to post their code if it's faster than mine. – user202729 May 31 '18 at 13:56
• Please?... (note: This is an instance of a set cover problem.) – user202729 Jun 1 '18 at 23:43
• I'm working on it. I just can't come up with any smart way of doing it. If nobody else posts an answer, I'll put mine up that can only go as high as n=4 so far. – mypetlion Jun 1 '18 at 23:58