Yet Another Trivial Binary Search Implementation
Difference between en19 and en20, changed 0 character(s)
These might be trivial for most users, but if you are like me, who sometimes confuse on when to use `mid+1`, `rig-1` etc, here is a reliable method after considering many different implementation variants.↵
↵
#### To find the index of the **rightmost $1$** in a monotonically decreasing function, for example $a=[1,1,1,1,0,0,0]$↵
↵
Define `check(i)` to return true when $a[i]=1$, false otherwise.↵
↵
~~~~~↵
int lef = 0, rig = n;↵
while(lef < rig) {↵
int mid=(lef + rig)/2;↵
if(check(mid)){↵
lef = mid+1;↵
} else {↵
rig = mid;↵
}↵
}↵
int ans = lef-1;↵
~~~~~↵
↵
Notice that the `mid` in `lef=mid+1` is always valid, so the answer (`lef-1`) is the last valid `mid`. Also notice the initial values of `lef` and `rig`. `rig` is set out of bounds. If $a$ is available or can be computed efficiently, you can instead do↵
↵
~~~~~↵
F0R(i,n)a[i]*=-1;↵
int ans = upper_bound(a,a+n,-1)-a-1; // get index of rightmost 1↵
~~~~~↵
Careful if $1$ doesn't exist in $a$.↵
↵
#### To find the index of the **leftmost $1$** in a monotonically increasing function, for example  $a=[0,0,0,0,1,1,1]$↵
↵
In a similar way, ↵
↵
~~~~~↵
int lef = -1, rig = n-1;↵
while(lef < rig) {↵
int mid=(lef + rig + 1)/2;↵
if(check(mid)){↵
rig = mid-1;↵
} else {↵
lef = mid;↵
}↵
}↵
int ans = rig+1;↵
~~~~~↵
↵
Notice the ceiling in the definition of `mid`. This is to handle `rig-lef` equals one. Again notice the initial values for `lef` and `rig`. `lef` is set out of bounds. If $a$ is available or can be computed efficiently, you can instead do↵
↵
~~~~~↵
int ans = lower_bound(a,a+n,1)-a; // get index of leftmost 1↵
~~~~~↵
Careful if $1$ doesn't exist in $a$.↵
↵
You can then solve most problems by implementing your own definition of `check()`. Time complexity is the time complexity of your `check()` function times $\log n$. Thanks to [user:marvenlee,2021-02-19] for inspiring this blog.↵
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Usage examples:↵
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- Recent problem 1486C2 [submission:107901422]↵
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- 367C [submission:107906435]↵
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- 604B [submission:107908687]↵
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- 237C [submission:107914994]↵
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Indeed, there are many other ways of implementing binary search, check out [pllk's blog](https://mirror.codeforces.com/blog/entry/9901).

History

 
 
 
 
Revisions
 
 
  Rev. Lang. By When Δ Comment
en24 English jeqcho 2021-02-19 17:05:48 26
en23 English jeqcho 2021-02-19 16:59:06 46
en22 English jeqcho 2021-02-19 16:56:04 10 Tiny change: 'he answer (`lef-1`) is the la' -> 'he answer is the la'
en21 English jeqcho 2021-02-19 16:47:18 438 Applied improvements thanks to [user:sergei_popov]
en20 English jeqcho 2021-02-19 09:59:01 0 (published)
en19 English jeqcho 2021-02-19 09:58:24 33 Tiny change: '08687]\n\nIndee' -> '08687]\n\n- 237C [submission:107914994]\n\nIndee'
en18 English jeqcho 2021-02-19 09:55:01 73
en17 English jeqcho 2021-02-19 09:54:14 3
en16 English jeqcho 2021-02-19 09:53:34 4 Tiny change: 't problem C2)\n\n- [' -> 't problem 1486C2)\n\n- ['
en15 English jeqcho 2021-02-19 09:53:10 7 Tiny change: '107906435]\n\n- [sub' -> '107906435] (367C)\n\n- [sub'
en14 English jeqcho 2021-02-19 09:52:46 7 Tiny change: '107908687]\n\nIndeed' -> '107908687] (604B)\n\nIndeed'
en13 English jeqcho 2021-02-19 09:49:12 28 Tiny change: 'nd `rig`. If $a$ i' -> 'nd `rig`. `lef` is set out of bounds. If $a$ i'
en12 English jeqcho 2021-02-19 09:46:20 22
en11 English jeqcho 2021-02-19 09:43:39 1
en10 English jeqcho 2021-02-19 09:42:50 80
en9 English jeqcho 2021-02-19 09:33:43 30
en8 English jeqcho 2021-02-19 09:31:45 94 Tiny change: ',a+n,-1)-a; // get i' -> ',a+n,-1)-a-1; // get i'
en7 English jeqcho 2021-02-19 09:24:13 264 Tiny change: 'e. If `a` can be co' -> 'e. If `a` is available or can be co'
en6 English jeqcho 2021-02-19 09:16:14 8 Tiny change: '`mid+1`, `lef+1` etc.\n\' -> '`mid+1`, `rig-1` etc.\n\'
en5 English jeqcho 2021-02-19 08:24:52 258 Tiny change: '0, rig = n-1;\nwhile(l' -> '0, rig = n;\nwhile(l'
en4 English jeqcho 2021-02-19 08:04:27 316 Tiny change: ' $\log N$.' -> ' $\log N$. Thanks to [user:marvenlee] for inspiring this blog.'
en3 English jeqcho 2021-02-19 07:42:06 3
en2 English jeqcho 2021-02-19 07:40:07 1015 Tiny change: '$a=[1,1,1,1,0,0,0]$\n' -> '$a=[1,1,1,$**$1$**$,0,0,0]$\n'
en1 English jeqcho 2021-02-19 07:15:15 157 Initial revision (saved to drafts)