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## Chapter 1. Boost.Icl

### Joachim Faulhaber

Copyright © 2007-2010 Joachim Faulhaber

Copyright © 1999-2006 Cortex Software GmbH

Distributed under the Boost Software License, Version 1.0. (See accompanying file LICENSE_1_0.txt or copy at http://www.boost.org/LICENSE_1_0.txt)

## Introduction

| Note
C++03 support is deprecated in Boost.Icl 1.84 and will be removed in Boost.Icl 1.86.

Intervals are almost ubiquitous in software development. Yet they are very easily coded into user defined classes by a pair of numbers so they are only implicitly used most of the time. The meaning of an interval is simple. They represent all the elements between their lower and upper bound and thus a set. But unlike sets, intervals usually can not be added to a single new interval. If you want to add intervals to a collection of intervals that does still represent a set, you arrive at the idea of interval_sets provided by this library.

Interval containers of the ICL have been developed initially at Cortex Software GmbH to solve problems related to date and time interval computations in the context of a Hospital Information System. Time intervals with associated values like amount of invoice or set of therapies had to be manipulated in statistics, billing programs and therapy scheduling programs. So the ICL emerged out of those industrial use cases. It extracts generic code that helps to solve common problems from the date and time problem domain and can be beneficial in other fields as well.

One of the most advantageous aspects of interval containers is their very compact representation of sets and maps. Working with sets and maps of elements can be very inefficient, if in a given problem domain, elements are typically occurring in contiguous chunks. Besides a compact representation of associative containers, that can reduce the cost of space and time drastically, the ICL comes with a universal mechanism of aggregation, that allows to combine associated values in meaningful ways when intervals overlap on insertion.

For a condensed introduction and overview you may want to look at the presentation material on the ICL from BoostCon2009.

### Definition and Basic Example

The Interval Container Library (ICL) provides `intervals` and two kinds of interval containers: `interval_sets` and `interval_maps`.

- An `interval_set` is a set that is implemented as a set of intervals.
- An `interval_map` is a map that is implemented as a map of interval value pairs.

##### Two Aspects

`Interval_sets` and `interval_maps` expose two different aspects in their interfaces: (1) The functionality of a set or a map, which is the more abstract aspect. And (2) the functionality of a container of intervals which is the more specific and implementation related aspect. In practice both aspects are useful and are therefore supported.

The first aspect, that will be called fundamental aspect, is the more important one. It means that we can use an `interval_set` or `interval_map` like a set or map of elements. It exposes the same functions.

    interval_set<int> mySet;
    mySet.insert(42);
    bool has_answer = contains(mySet, 42);

The second aspect, that will be called segmental aspect, allows to exploit the fact, that the elements of `interval_sets` and `interval_maps` are clustered in intervals or segments that we can iterate over.

    // Switch on my favorite telecasts using an interval_set
    interval<seconds>::type news(make_seconds("20:00:00"), make_seconds("20:15:00"));
    interval<seconds>::type talk_show(make_seconds("22:45:30"), make_seconds("23:30:50"));
    interval_set<seconds> myTvProgram;
    myTvProgram.add(news).add(talk_show);

    // Iterating over elements (seconds) would be silly ...
    for(interval_set<seconds>::iterator telecast = myTvProgram.begin();
        telecast != myTvProgram.end(); ++telecast)
        //...so this iterates over intervals
       TV.switch_on(*telecast);

Working with `interval_sets` and `interval_maps` can be beneficial whenever the elements of sets appear in contiguous chunks: `intervals`. This is obviously the case in many problem domains, particularly in fields that deal with computations related to date and time.

##### Addabitlity and Subtractability

Unlike `std::sets` and `maps`, `interval_sets` and `interval_maps` implement concept `Addable` and `Subtractable`. So `interval_sets` define an `operator +=` that is naturally implemented as set union and an `operator -=` that is consequently implemented as set difference. In the Icl `interval_maps` are addable and subtractable as well. It turned out to be a very fruitful concept to propagate the addition or subtraction to the `interval_map's` associated values in cases where the insertion of an interval value pair into an `interval_map` resulted in a collision of the inserted interval value pair with interval value pairs, that are already in the `interval_map`. This operation propagation is called aggregate on overlap.

##### Aggregate on Overlap

This is a first motivating example of a very small party, demonstrating the aggregate on overlap principle on `interval_maps`:

In the example Mary enters the party first. She attends during the time interval `[20:00,22:00)`. Harry enters later. He stays within `[21:00,23:00)`.

    typedef std::set<string> guests;
    interval_map<time, guests> party;
    party += make_pair(interval<time>::right_open(time("20:00"), time("22:00")), guests("Mary"));
    party += make_pair(interval<time>::right_open(time("21:00"), time("23:00")), guests("Harry"));
    // party now contains
    [20:00, 21:00)->{"Mary"}
    [21:00, 22:00)->{"Harry","Mary"} //guest sets aggregated on overlap
    [22:00, 23:00)->{"Harry"}

On overlap of intervals, the corresponding name sets are accumulated. At the points of overlap the intervals are split. The accumulation of content on overlap of intervals is done via an `operator +=` that has to be implemented for the associated values of the `interval_map`. In the example the associated values are `guest sets`. Thus a `guest set` has to implement `operator +=` as set union.

As can be seen from the example an `interval_map` has both a decompositional behavior (on the time dimension) as well as an accumulative one (on the associated values).

Addability and aggregate on overlap are useful features on `interval_maps` implemented via function `add` and `operator +=`. But you can also use them with the traditional insert semantics that behaves like `std::map::insert` generalized for interval insertion.

### Icl's class templates

In addition to interval containers we can work with containers of elements that are behavioral equal to the interval containers: On the fundamental aspect they have exactly the same functionality. An `std::set` of the STL is such an equivalent set implementation. Due to the aggregation facilities of the icl's interval maps `std::map` is fundamentally not completely equivalent to an `interval_map`. Therefore there is an extra `icl::map` class template for maps of elements in the icl.

- The `std::set` is behavioral equal to `interval_sets` on the fundamental aspect.
- An `icl::map` is behavioral equal to `interval_maps` on the fundamental aspect. Specifically an `icl::map` implements aggregate on overlap, which is named aggregate on collision for an element container.

The following tables give an overview over the main class templates provided by the icl.

Table 1.1. Interval class templates

group | form | template
--- | --- | ---
statically bounded | asymmetric | `right_open_interval`
|  | `left_open_interval`
| symmetric | `closed_interval`
|  | `open_interval`
dynamically bounded |  | `discrete_interval`
|  | `continuous_interval`

Statically bounded intervals always have the same kind of interval borders, e.g. right open borders`[a..b)` for `right_open_interval`. Dynamically bounded intervals can have different borders. Refer to the chapter about intervals for details.

Table 1.2. Container class templates

granularity | style | sets | maps
--- | --- | --- | ---
interval | joining | `interval_set` | `interval_map`
| separating | `separate_interval_set` |
| splitting | `split_interval_set` | `split_interval_map`
element |  | (`std::set` ) | `map`

`Std::set` is placed in paretheses, because it is not a class template of the ICL. It can be used as element container though that is behavioral equal to the ICL's interval sets on their fundamental aspect. Column style refers to the different ways in which interval containers combine added intervals. These combining styles are described in the next section.

### Interval Combining Styles

When we add intervals or interval value pairs to interval containers, the intervals can be added in different ways: Intervals can be joined or split or kept separate. The different interval combining styles are shown by example in the tables below.

Table 1.3. Interval container's ways to combine intervals

| joining | separating | splitting
--- | --- | --- | ---
set | `interval_set` | `separate_interval_set` | `split_interval_set`
map | `interval_map` |  | `split_interval_map`
| Intervals are joined on overlap or touch (if associated values are equal). | Intervals are joined on overlap, not on touch. | Intervals are split on overlap. All interval borders are preserved.

Table 1.4. Interval combining styles by example

| joining | separating | splitting
--- | --- | --- | ---
set | `interval_set` A | `separate_interval_set` B | `split_interval_set` C
      {[1      3)          }
    +       [2      4)
    +                 [4 5)
    = {[1                5)}
      {[1      3)}         }
    +       [2      4)
    +                 [4 5)
    = {[1           4)[4 5)}
      {[1      3)          }
    +       [2      4)
    +                 [4 5)
    = {[1 2)[2 3)[3 4)[4 5)}
map | `interval_map` D |  | `split_interval_map` E
      {[1      3)->1          }
    +       [2      4)->1
    +                 [4 5)->1
    = {[1 2)[2 3)[3      5)   }
      | ->1  ->2     ->1      |
      {[1      3)->1          }
    +       [2      4)->1
    +                 [4 5)->1
    = {[1 2)[2 3)[3 4)[4 5)   }
      | ->1  ->2  ->1  ->1    |

Note that `interval_sets` A, B and C represent the same set of elements `{1,2,3,4}` and `interval_maps` D and E represent the same map of elements `{1->1, 2->2, 3->1, 4->1}`. See example program Interval container for an additional demo.

##### Joining interval containers

`Interval_set` and `interval_map` are always in a minimal representation. This is useful in many cases, where the points of insertion or intersection of intervals are not relevant. So in most instances `interval_set` and `interval_map` will be the first choice for an interval container.

##### Splitting interval containers

`Split_interval_set` and `split_interval_map` on the contrary have an insertion memory. They do accumulate interval borders both from additions and intersections. This is specifically useful, if we want to enrich an interval container with certain time grids, like e.g. months or calendar weeks or both. See example time grids for months and weeks.

##### Separating interval containers

`Separate_interval_set` implements the separating style. This style preserves borders, that are never passed by an overlapping interval. So if all intervals that are inserted into a `separate_interval_set` are generated form a certain grid that never pass say month borders, then these borders are preserved in the `separate_interval_set`.

14:46 15.10.2010