Re: Number of interval Intersections for a large number of

• To: mathgroup at smc.vnet.net
• Subject: [mg81832] Re: Number of interval Intersections for a large number of
• From: Ray Koopman <koopman at sfu.ca>
• Date: Thu, 4 Oct 2007 04:29:57 -0400 (EDT)
• References: <fdt3r8\$s2i\$1@smc.vnet.net>

```On Oct 2, 2:43 am, P_ter <peter_van_summe... at yahoo.co.uk> wrote:
> It seems I was not clear enough in my problem.
> I have a few thousand intervals {e1,e2,...} each of which is like {begin,end}
> The elements e are ordered according to their begin value.
> Some intervals do not intersect. But it also can be like this (the points are only for making a clear drawing):
> .......<------>
> ...<------------------->
> <------------------>
> So, in this case the first interval can be partioned :
>       in a part which has no intersection with the other two,
>       in a part which intersects twice,
>       in a part which intersects three times,
>       in a part which intersects two times.
> For each interval I make a Block[x] = UnitStep[x-beginvalue]-UnitStep[x-endvalue]
> I sum all the Blocks to one function and ask Reduce to find me a solution for those parts of intervals which intersects only three times: Reduce[SumOfBlocks[x]==3,x].
> Because it is a sum of UnitSteps and where there are three times an intersection, the value is 3.
> And so on for 1,2,4,5,etc.
> It is a safe way to do it, but not so fast. Are there other algorithms? Where can I find them?

Here's a slightly simpler way to do it.

Generate some intervals.

In[1]:=
t = Sort[Sort/@Table[Random[],{n = 10},{2}]]*(max = 100)

Out[1]=
{{2.91777,91.5483},{6.76612,16.9744},{10.6363,48.1385},
{14.2374,41.9963},{14.4536,73.1989},{17.7657,19.4148},
{20.027,32.2291},{28.9051,29.0713},{50.0155,67.9657},
{65.3366,78.6108}}

In[2]:=

Get the overlaps.

In[3]:=
{p,q} = {Partition[ Join[{0},#[[All,1]],{max}], 2, 1],
FoldList[ Plus, 0, #[[All,2]]]}& @
Sort@Flatten[Transpose@{#,{1,-1}}&/@t, 1]
r = {#,Pick[p,q,#]}& /@ Range[0,Max@q]

Out[3]=
{{{0,2.91777},{2.91777,6.76612},{6.76612,10.6363},
{10.6363,14.2374},{14.2374,14.4536},
{14.4536,16.9744},{16.9744,17.7657},
{17.7657,19.4148},{19.4148,20.027},
{20.027,28.9051},{28.9051,29.0713},
{29.0713,32.2291},{32.2291,41.9963},
{41.9963,48.1385},{48.1385,50.0155},
{50.0155,65.3366},{65.3366,67.9657},
{67.9657,73.1989},{73.1989,78.6108},
{78.6108,91.5483},{91.5483,100}},
{0,1,2,3,4,5,4,5,4,5,6,5,4,3,2,3,4,3,2,1,0}}

Out[4]=
{{0,{{0,2.91777},{91.5483,100}}},
{1,{{2.91777,6.76612},{78.6108,91.5483}}},
{2,{{6.76612,10.6363},{48.1385,50.0155},{73.1989,78.6108}}},
{3,{{10.6363,14.2374},{41.9963,48.1385},{50.0155,65.3366},
{67.9657,73.1989}}},
{4,{{14.2374,14.4536},{16.9744,17.7657},{19.4148,20.027},
{32.2291,41.9963},{65.3366,67.9657}}},
{5,{{14.4536,16.9744},{17.7657,19.4148},{20.027,28.9051},
{29.0713,32.2291}}},
{6,{{28.9051,29.0713}}}}

In[5]:=

If you don't want all of r at once, Pick[p,q,x] will give
a list of the intervals whose overlap depth is x.

```

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