# Is it a prefix normal word?

Prefix normal words arise in the context of binary jumbled pattern matching. A binary word $$\w\$$ consisting of $$\0\$$s and $$\1\$$s is said to be prefix normal * if, among all of its substrings, none contains more $$\1\$$s than the prefix of $$\w\$$ of the same length does. In other words, if $$\w\$$ contains $$\n\$$ $$\1\$$s then it is prefix normal if, for all $$\m\le n\$$, the shortest (or equal shortest) substring containing $$\m\$$ $$\1\$$s is a prefix.

For example, $$\w=\color{red}{11010}\color{blue}{11011}00100\$$ is not prefix normal because it has a substring of length 5 (highlighted in blue) that contains four $$\1\$$s, whereas its prefix of length 5 (red) only contains three $$\1\$$s. If we flip the first $$\0\$$ to $$\1\$$, however, then the resulting word ($$\111101101100100\$$) is prefix normal.

Your task in this challenge is to write a program or function that decides whether a binary word is prefix normal.

You may take input in any sensible format (e.g. string, numeric, list of characters/digits), which extends to using any pair of characters/digits instead of $$\0\$$ and $$\1\$$ if you wish.

Output/return a consistent value for every input that is prefix normal and another consistent value for every input that is not.

### Test cases

Prefix normal

00000
1101010110
111101101100100
11101010110110011000
1110010110100111001000011


Not prefix normal

00001
1011011000
110101101100100
11010100001100001101
1110100100110101010010111


* Strictly, prefix normal with respect to $$\1\$$.

# Jelly, 9 7 bytes

ṡJ§Ṁ€⁼Ä


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-2 bytes thanks to Unrelated String

ṡJ§Ṁ€⁼Ä  Main Link
ṡ        Take overlapping slices of length(s)
J       [1, 2, ..., length]
§      Take the sum of each slice of each length
Ṁ€    Is the maximum sum for each length
⁼   Equal to
Ä  The cumulative sum of the original list?

• Looks like it beats mine if you borrow Ä from it... which is unfortunate because it really feels like there should be some Ƒ trick you can pull with »\  or something Oct 17, 2021 at 4:26
• (that ṡJ is super nice by the way) Oct 17, 2021 at 4:34
• @UnrelatedString Oh yeah I didn't realize that the first of each sublist is literally just the prefix sums lol. thanks Oct 17, 2021 at 7:23

g x=x!x
p!s=p==[]||sum p>=sum s&&init p!tail s&&g(init p)


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A bit longer, but actually runs in reasonable time.

g x|k<-[1..length x]=and[h x>=h(drop n x)|h<-(sum.).take<\$>k,n<-k]


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# JavaScript (ES6), 57 bytes

Expects an array of binary digits. Returns false for prefix normal, or true for not prefix normal.

a=>a.map(v=>a=v-~a).some((p,i,a)=>a.some(q=>q-a[~i--]>p))


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### How?

We first compute the cumulative sums of the incremented input values:

a.map(v => a = v - ~a)


(Using a = v - ~a allows us to re-use a[] to compute the sum, which is two bytes shorter than allocating a specific variable. That's the only reason why the values are incremented.)

For instance:

   1, 0, 1, 1, 0, 1, 1, 0, 0, 0
=> 2, 1, 2, 2, 1, 2, 2, 1, 1, 1
=> 2, 3, 5, 7, 8,10,12,13,14,15


We then test whether there's some pair $$\(i,j)\$$ with $$\j>i\$$ such that $$\a_j-a_{j-i-1}>a_i\$$.

.some((p, i, a) => a.some(q => q - a[~i--] > p))
//                             ^      ^      ^
//                           a[j]  a[j-i-1]  a[i]


# Wolfram Language (Mathematica), 54...44 41 bytes

(l=Most@l+#-(t={##2})&@@t)&/@(t=l=#)0&


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 is VectorGreaterEqual.

(l=Most@l+#-(t={##2})&@@t)&/@(t=l=#)        get all prefix differences by:
(t=l=#)          l: difference list, t: input list
/@                 over input length:
(l=Most@l                )&                     drop last element of l
+#-   {##2} &@@t                       foreach, add next in prefix, subtract next in substring
e.g. (a+b+c)-(b+c+d) -> (a+b+c)-(b+c+d)+d-e
(t=     )                           shift t left
0      all nonnegative?


*strictly speaking, each element of l is also offset by corresponding input element . I suspect this doesn't matter - a brute force test of strings up to length 17 showed no differences - but I don't have a proof as to why.
If it does make a difference, +1 byte using (l=Most@l+#-{t=##2}&@t)&/@(l=0{t=##})0& instead. Try it online!

# Jelly, 9 8 bytes

ẆẈṀƙ§Ɗ⁼Ä


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It feels like something less naive should be shorter yet, but I haven't had any luck.

Ẇ           Get every substring of the input.
§       Sum each substring,
ƙ Ɗ      group the sums by
Ẉ          the lengths of the corresponding substrings,
Ṁ         and take the largest element of each group.
⁼Ä    Is the resulting list equal to the cumulative sums of the input?


# Risky, 31 bytes

*00_{?*_1_+!?{_?+0_{?*_1_+_?*_1__[___{_0*__1*_1:____{_0{__1*_1


Input is a single argument which is a list of 0's and 1's; output is 1 if prefix normal, 0 if not. Try it online!

### Explanation

This is a bit long, so I'll explain it in two halves.

00_{?*_1_+!?{_?+0_{?*_1_+_?*_1
?                           Input list
{                            List of all sublists
*_1                        Repeat once (no-op)
_                             Same value (no-op)
_                       Map this function to each of the sublists
(first arg is each sublist, second arg is original input):
!?                     Length of first arg
{                    Take prefix of that length from
_?                  Second arg
+                       Sum
0                              Transpose (adds a level of nesting to a flat list)
+                Concatenate that list with the following:
{?             List of all sublists of input list
*_1          Repeat once (no-op)
_               Same value (no-op)
_         Map this function to each sublist:
_?       First arg
*_1    Repeat 1 time (no-op)
+         Sum
0                               Transpose


At this point, we have a list of pairs $$\(n_l^0, n_l^i)\$$, where $$\n_l^i\$$ is the number of ones in the $$\i^{th}\$$ sublist of length $$\l\$$ and $$\n_l^0\$$ is the number of ones in the first sublist of length $$\l\$$. We now want to check if for every pair, $$\n_l^0 \geq n_l^i\$$. Equivalently, $$\min(n_l^0, n_l^i) = n_l^i\$$.

__[___{_0*__1*_1:____{_0{__1*_1
_                                Map this function to each pair (first arg is each pair,
second arg is original input):
__                            List of both args
{_0                         Get 0th element (the pair)
_                              Same value (no-op)
*__1*_1                  Repeat 1 time (no-op)
[                               Minimum
:                 Equals
__              List of both args
{_0           Get 0th element (the pair)
_                Same value (no-op)
{          Get element of the pair at index
__1*_1    1 (the second element)
_                 Same value (no-op)
_                                Same value (no-op)


We now have a list containing 0 for every sublist with more ones than the equal-length prefix, and 1 for every other sublist. The * at the beginning of the code takes the product of this list, yielding 0 for inputs with any such substrings and 1 for inputs without any.

# Retina 0.8.2, 49 bytes

((1)|0)+(?<=(?!(?>(?<-1>(?<-2>1)|0)*)(?(2)^))^.*)


Try it online! Outputs 0 for prefix normal, 1 for not normal. Explanation:

((1)|0)+


Match some 1s and 0s, keeping count of the 1s and the total number of digits (very slightly golfier as we need a group for the +), ...

(?<=...^.*)


... then, at the beginning of the string, ...

(?>(?<-1>(?<-2>1)|0)*)


... try to consume all of the 1s, matching up to the same number of digits...

(?!...(?(2)^))


... and fail if there were enough 1s, showing that the prefix is normal.

Note that if there are two disjoint substrings with too many 1s then I think the substring containing both substrings will also have too many 1s and will therefore be found first, but I can't prove that.

# 05AB1E, 9 bytes

Œε∍OyO@}P


Input as a list of 0s/1s.

Explanation:

Œ        # Get all sublists of the (implicit) input-list
ε       # Map over each sublist:
∍      #  Shorten the (implicit) input-list to a length equal to this sublist-length
O     #  Sum this prefix
yO   #  Sum the sublist as well
@  #  Check that the prefix-sum is larger than or equal to the sublist-sum
}P      # After the map: pop and take the product to check if all were truthy
# (after which it is output implicitly as result)


# APL (Dyalog Unicode), 14 bytes

∧/∘∊+\≥⍳∘≢+/¨⊂


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A tacit function taking a vector of 0's and 1's

⊂ The input vector enclosed in a singleton list.
⍳∘≢ indices of the input (1 .. length input)
¨ For each of the numbers on the left and the input vector on the right:
+/ sums of all sublists of the right argument of the length given by the left argument.
+\ sums of prefixes of the input.
≥ for each prefix sum, is it larger or equal than each of the sublist sum of sublists of the same size.
∊ flatten nested list.
∧/ reduce with boolean and.

Example with input 1 0 0 1 1:

┌──┬─────────┬──────────┐
│+\│⍳∘≢+/¨⊂  │+\≥⍳∘≢+/¨⊂│
├──┼─────────┼──────────┤
│1 │1 0 0 1 1│1 1 1 1 1 │
├──┼─────────┼──────────┤
│1 │1 0 1 2  │1 1 1 0   │
├──┼─────────┼──────────┤
│1 │1 1 2    │1 1 0     │
├──┼─────────┼──────────┤
│2 │2 2      │1 1       │
├──┼─────────┼──────────┤
│3 │3        │1         │
└──┴─────────┴──────────┘


# Scala, 74 bytes

s=>1 to s.size forall(l=>s.sliding(l).map(_.sum).forall(_<=s.take(l).sum))


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Strategy is as follows:

• for all possible prefix lengths (1 to s.size forall{...}) check the following
• compute the number of 1's within each sliding window of current prefix length (s.sliding(l).map(_.sum))
• check whether all these sliding windows contain no more 1's than the first window/initial prefix (.forall(_<=s.take(l).sum))

# Ruby, 70 60 bytes

->s,r=0..s.size{r.any?{|i|r.any?{|j|s[j,i].sum>s[0,i].sum}}}


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• Saved 9 Bytes thanks to @Dingus suggestions !
any? replaces first map
We reuse entire range L=0..s.size instead of just the length

• Saved another 1 again by @Dingus
Compares directly the sums of prefix and current range being checked

• Takes an array s of 0/1 and return false if it is prefix normal, true if not.

(1..L=s.size).map{|i|  - every length by*
(X= ... )        - we save in X:
(0..L).map{|j|   * every index: all substrings of same length
s[j,i].sum}      * sum 1's in slice of length i at j
.max>X[0]     - any > of 1st element?
}.any?}          - any true?

• Couple of hints towards a 60 byte solution: both loops can be over the same range, and any? can take a block (so map can be avoided). Oct 18, 2021 at 1:30
• Here is my 60 :) Oct 18, 2021 at 10:33
• @Dingus I was just wondering how was your 60 bytes Oct 18, 2021 at 11:52

# R, 92 bytes

Or R>=4.1, 78 bytes by replacing two function appearances with \s.

function(s,n=sum(s|1)){for(i in 1:n)F=F|which.max(sapply(0:n,function(j)sum(s[1:i+j])))-1;F}


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Takes a vector of integers. Outputs FALSE for prefix normal and TRUE for non-prefix normal word.

Solution shorter in R>=4.1:

# R, 98 bytes

Or R>=4.1, 77 bytes by replacing three function appearances with \s.

function(s,n=sum(s|1),g=sapply)any(g(1:n,function(i)which.max(g(0:n,function(j)sum(s[1:i+j])))-1))


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# Charcoal, 19 18 bytes

⊙θΦκ‹Σ…θ⁻⊕κλΣ✂θλ⊕κ


Try it online! Link is to verbose version of code. Outputs - for not normal, nothing for prefix normal. Explanation:

 θ                  Input string
⊙ Φκ                Any substring satisfies
Σ              Sum of digits
‹               Less than
✂θλ⊕κ  Current substring
Σ       Sum of digits
Implicitly print

• @Dingus Although, that got me thinking, and I came up with a way to save a byte, so that's this cloud's silver lining!
– Neil
Oct 17, 2021 at 13:37

# Core Maude, 186 bytes

mod P is pr NAT-LIST . var L P F X Y Z :[Nat]. ops n c : Nat ~> Nat . ceq
n(L)= F if P X 2 Y F Z := L 2 L /\ size(P)= size(F)/\ c(F)> c(P). eq c(X 1 Y)=
s c(X Y). eq c(L)= 0[owise]. endm


The answer is obtained by reducing the n function with the input bitstring as a Maude NatList of 0s and 1s. If the term is irreducible (result is an “error expression” of kind [NatList]), then the bitstring is prefix normal, otherwise (result of sort NeNatList, Nat, etc.) it is not.

### Example Session

             \||||||||||||||||||/
--- Welcome to Maude ---
/||||||||||||||||||\
Maude 3.1 built: Oct 12 2020 20:12:31
Sun Oct 17 22:18:44 2021
Maude> mod P is pr NAT-LIST . var L P F X Y Z :[Nat]. ops n c : Nat ~> Nat . ceq
> n(L)= F if P X 2 Y F Z := L 2 L /\ size(P)= size(F)/\ c(F)> c(P). eq c(X 1 Y)=
> s c(X Y). eq c(L)= 0[owise]. endm
Maude> red n(0 0 0 0) .
result [NatList]: n(0 0 0 0)
Maude> red n(1 1 0 1 0 1 0 1 1 0) .
result [NatList]: n(1 1 0 1 0 1 0 1 1 0)
Maude> red n(1 1 1 1 0 1 1 0 1 1 0 0 1 0 0) .
result [NatList]: n(1 1 1 1 0 1 1 0 1 1 0 0 1 0 0)
Maude> red n(1 1 1 0 1 0 1 0 1 1 0 1 1 0 0 1 1 0 0 0) .
result [NatList]: n(1 1 1 0 1 0 1 0 1 1 0 1 1 0 0 1 1 0 0 0)
Maude> red n(1 1 1 0 0 1 0 1 1 0 1 0 0 1 1 1 0 0 1 0 0 0 0 1 1) .
result [NatList]: n(1 1 1 0 0 1 0 1 1 0 1 0 0 1 1 1 0 0 1 0 0 0 0 1 1)
Maude> red n(0 0 0 0 1) .
result NzNat: 1
Maude> red n(1 0 1 1 0 1 1 0 0 0) .
result NeNatList: 1 1
Maude> red n(1 1 0 1 0 1 1 0 1 1 0 0 1 0 0) .
result NeNatList: 1 1 0 1 1
Maude> red n(1 1 0 1 0 1 0 0 0 0 1 1 0 0 0 0 1 1 0 1) .
result NeNatList: 1 1 0 0 0 0 1 1 0 1
Maude> red n(1 1 1 0 1 0 0 1 0 0 1 1 0 1 0 1 0 1 0 0 1 0 1 1 1) .
result NeNatList: 1 0 1 0 0 1 0 1 1 1
Maude> quit
Bye.


### Ungolfed

mod P is
pr NAT-LIST .
var L P F X Y Z : [Nat] .
ops n c : Nat ~> Nat .
ceq n(L) = F if P X 2 Y F Z := L 2 L /\ size(P) = size(F) /\ c(F) > c(P) .
eq c(X 1 Y) = s c(X Y) .
eq c(L) = 0 [owise] .
endm


We use Maude's strong pattern matching and conditional equations to try every possible prefix and sublist. The c function counts the number of 1s in a list.

Interesting note: If the input term is reducible, the result will be a witness sublist with more 1s than the prefix of the same length. It did not cost any extra bytes to do so.

The [owise] attribute does not seem to be needed in the current Maude 3.1 interpreter (the interpreter seems to prefer earlier equations over later ones), but Maude's definition of an “admissible functional module” requires it. Otherwise we could save 6 bytes by omitting it.

• This looks like an interesting (and somewhat specialised) language. Is the output not always the same for the non-normal cases? (A consistent value is required.) Oct 18, 2021 at 11:01
• I missed the bit in your question about a consistent value. Currently I "return a boolean value via the presence or absence of an error/exception" at described in the FAQ: codegolf.meta.stackexchange.com/a/11908/45870 Oct 19, 2021 at 8:39
• Does that count as consistent? I'm returning true or false, just encoded. Oct 19, 2021 at 9:00
• The query I have is that the output looks to be different for every non-normal case. However, I'm not sure how much control you have over it, i.e. what is printed automatically by the interpreter and what isn't. If all the output is automatic I'd consider it acceptable. Oct 19, 2021 at 11:28
• The interpreter always prints the result of evaluation. To print a constant for the non-normal case would be easy (change F to, e.g., 0), but for the normal case I would have to make the function total to produce a constant. In Maude, evaluating to an error term (with no sort) is roughly equivalent to an exception, and evaluating to a normal term (with a sort) is successful execution. My thought was the result doesn't really matter if I'm using "error" and "no error" to encode false and true. Oct 21, 2021 at 0:13

# Python 3, 91 79 bytes

lambda s:any(sum(s[:i-j])<sum(s[j:i])for i in range(len(s)+1)for j in range(i))


Take as input a list of integers.

Return False for prefix normal and True for prefix not normal

Thanks @UnrelatedString for -12 bytes using list of integers instead of strings

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