# Interval Indices

## Interval index basics

`interval_index` is a persistent structure that associates elements with interval endpoints and supports fast interval-relation queries. Intervals are stored sorted by start endpoint, with stable first-in-first-out (FIFO) ordering for ties.

All operations are persistent, and return modified copies. Peeking returns the stored element (which may be any type), popping returns a list with `$value` containing the stored element, `$start` and `$end` the interval endpoints, and `$remaining` the rest of the index without that element.

    ix <- interval_index(
      "A", "B", "C",
      start = c(1, 2, 4),
      end = c(3, 4, 5)
    )
    ix
    #> Unnamed interval_index with 3 elements, default query bounds [start, end).
    #>
    #> Elements (by interval start order):
    #>
    #> [[1]] (interval 1 - 3)
    #> [1] "A"
    #>
    #> [[2]] (interval 2 - 4)
    #> [1] "B"
    #>
    #> [[3]] (interval 4 - 5)
    #> [1] "C"

The `as_interval_index()` variant builds an index from a vector or list of elements paired with start/end vectors, useful when endpoints are already in separate vectors.

    ix2 <- as_interval_index(c("phase1", "phase2", "phase3"), start = c(1, 3, 2), end = c(4, 5, 6))
    ix2
    #> Unnamed interval_index with 3 elements, default query bounds [start, end).
    #>
    #> Elements (by interval start order):
    #>
    #> [[1]] (interval 1 - 4)
    #> [1] "phase1"
    #>
    #> [[2]] (interval 2 - 6)
    #> [1] "phase3"
    #>
    #> [[3]] (interval 3 - 5)
    #> [1] "phase2"

`as_flexseq(ix)` returns a payload-only `flexseq`; interval bounds and any custom monoids are dropped (see `as.list` to convert and preserve interval metadata).

## Point and interval queries

`peek_point()` returns the first element (in canonical interval order — smallest start, with insertion/FIFO order breaking ties among equal starts) whose interval contains a query point; `peek_all_point()` returns all matches as an `interval_index` slice.

    peek_point(ix, point = 2)
    #> [1] "A"
    peek_all_point(ix, point = 2)
    #> Unnamed interval_index with 2 elements, default query bounds [start, end).
    #>
    #> Elements (by interval start order):
    #>
    #> [[1]] (interval 1 - 3)
    #> [1] "A"
    #>
    #> [[2]] (interval 2 - 4)
    #> [1] "B"

`pop_point()` removes and returns the first match as `$value`/`$start`/`$end`/`$remaining`. Its “all” counterpart `pop_all_point()` returns `$elements` (an interval_index of removed matches) and `$remaining`.

    res <- pop_point(ix, 2)
    res$value
    #> [1] "A"
    res$start
    #> [1] 1
    res$end
    #> [1] 3
    res$remaining
    #> Unnamed interval_index with 2 elements, default query bounds [start, end).
    #>
    #> Elements (by interval start order):
    #>
    #> [[1]] (interval 2 - 4)
    #> [1] "B"
    #>
    #> [[2]] (interval 4 - 5)
    #> [1] "C"

By default the four `*_point()` functions test interval containment. The `match_at` parameter lets them match instead on coordinate equality at the entry’s start, end, or either endpoint. This is useful for sweep-line patterns where you need to identify intervals arriving or retiring at a specific event point. `bounds` is ignored for these modes (coordinate equality is structural).

    # Entries whose start coordinate equals 2
    peek_all_point(ix, point = 2, match_at = "start")
    #> Unnamed interval_index with 1 element, default query bounds [start, end).
    #>
    #> Elements (by interval start order):
    #>
    #> [[1]] (interval 2 - 4)
    #> [1] "B"

    # Entries whose end coordinate equals 3 — the sweep-line retirement query
    peek_all_point(ix, point = 3, match_at = "end")
    #> Unnamed interval_index with 1 element, default query bounds [start, end).
    #>
    #> Elements (by interval start order):
    #>
    #> [[1]] (interval 1 - 3)
    #> [1] "A"

    # Either endpoint equals 3
    peek_all_point(ix, point = 3, match_at = "either")
    #> Unnamed interval_index with 1 element, default query bounds [start, end).
    #>
    #> Elements (by interval start order):
    #>
    #> [[1]] (interval 1 - 3)
    #> [1] "A"

Interval-vs-interval queries select overlaps of different kinds. First we’ll build a new index to query:

    ix3 <- interval_index(
      "narrow", "wide", "right", "inner",
      start = c(2, 1, 4, 3),
      end   = c(3, 6, 5, 4)
    )
    ix3
    #> Unnamed interval_index with 4 elements, default query bounds [start, end).
    #>
    #> Elements (by interval start order):
    #>
    #> [[1]] (interval 1 - 6)
    #> [1] "wide"
    #>
    #> [[2]] (interval 2 - 3)
    #> [1] "narrow"
    #>
    #> [[3]] (interval 3 - 4)
    #> [1] "inner"
    #>
    #> [[4]] (interval 4 - 5)
    #> [1] "right"

`peek_overlaps()` returns any interval that shares at least one point with the query range. `peek_containing()` returns intervals that fully contain the query range. `peek_within()` returns intervals fully contained by the query range.

    # overlaps [2, 5]: any shared point
    peek_all_overlaps(ix3, start = 2, end = 5)
    #> Unnamed interval_index with 4 elements, default query bounds [start, end).
    #>
    #> Elements (by interval start order):
    #>
    #> [[1]] (interval 1 - 6)
    #> [1] "wide"
    #>
    #> [[2]] (interval 2 - 3)
    #> [1] "narrow"
    #>
    #> [[3]] (interval 3 - 4)
    #> [1] "inner"
    #>
    #> [[4]] (interval 4 - 5)
    #> [1] "right"

    # containing [3, 4]: index interval must enclose query
    peek_all_containing(ix3, start = 3, end = 4)
    #> Unnamed interval_index with 2 elements, default query bounds [start, end).
    #>
    #> Elements (by interval start order):
    #>
    #> [[1]] (interval 1 - 6)
    #> [1] "wide"
    #>
    #> [[2]] (interval 3 - 4)
    #> [1] "inner"

    # within [1, 5]: index interval must fit inside query
    peek_all_within(ix3, start = 1, end = 5)
    #> Unnamed interval_index with 3 elements, default query bounds [start, end).
    #>
    #> Elements (by interval start order):
    #>
    #> [[1]] (interval 2 - 3)
    #> [1] "narrow"
    #>
    #> [[2]] (interval 3 - 4)
    #> [1] "inner"
    #>
    #> [[3]] (interval 4 - 5)
    #> [1] "right"

All relation queries have symmetric `pop_*` and `pop_all_*` variants following the same return contract as `pop_point()` and `pop_all_point()`.

## Boundary modes

Endpoint inclusion is controlled by two related parameters. The constructors take `default_query_bounds` — one of `"[)"` (default, left-closed right-open), `"[]"` (closed), `"()"` (open), or `"(]"` (left-open right-closed) — which sets the convention used when queries on this index don’t specify their own. Query functions (`peek_*` / `pop_*`) then accept a `bounds` override for the duration of a single call.

    edge <- interval_index("E", start = 1, end = 3, default_query_bounds = "[)")

    # right endpoint excluded by default
    peek_point(edge, 3)
    #> NULL

    # override to closed bounds for this query
    peek_point(edge, 3, bounds = "[]")
    #> [1] "E"

Note that bounds affect what counts as a valid interval: under `"[)"`, an interval with `start == end` like [2, 2) is empty and will never match any query. Use `"[]"` if you need point intervals where `start == end`.

    # [2, 2) contains nothing under half-open bounds
    pt_open <- interval_index("X", start = 2, end = 2, default_query_bounds = "[)")
    peek_point(pt_open, point = 2)
    #> NULL

    # [2, 2] contains exactly the point 2
    pt_closed <- interval_index("X", start = 2, end = 2, default_query_bounds = "[]")
    peek_point(pt_closed, point = 2)
    #> [1] "X"

## Insertion and endpoint extrema

`insert()` adds an element preserving interval start order.

    ix4 <- insert(ix, "D", start = 2, end = 6)
    ix4
    #> Unnamed interval_index with 4 elements, default query bounds [start, end).
    #>
    #> Elements (by interval start order):
    #>
    #> [[1]] (interval 1 - 3)
    #> [1] "A"
    #>
    #> [[2]] (interval 2 - 4)
    #> [1] "B"
    #>
    #> [[3]] (interval 2 - 6)
    #> [1] "D"
    #>
    #> [[4]] (interval 4 - 5)
    #> [1] "C"

`min_endpoint()` and `max_endpoint()` return the current smallest start and largest end endpoints (not the stored elements).

    min_endpoint(ix)
    #> [1] 1
    max_endpoint(ix)
    #> [1] 5
    min_endpoint(interval_index())
    #> NULL

## Empty indices

Query and endpoint helpers are non-throwing on empty indices, and `length()` allows for empty checking.

    empty_ix <- interval_index()
    length(empty_ix)
    #> [1] 0
    peek_point(empty_ix, point = 1)
    #> NULL
    pop_overlaps(empty_ix, start = 1, end = 5)
    #> $value
    #> NULL
    #>
    #> $start
    #> NULL
    #>
    #> $end
    #> NULL
    #>
    #> $remaining
    #> Unnamed interval_index with 0 elements, default query bounds [start, end).

## Named interval indices

Interval indices can carry names, set either at construction or through `as_interval_index()` on a named list. Names support read-only indexing via `[`, `[[`, and `$`; all replacement forms (`[<-`, `[[<-`, `$<-`) error, because index-based assignment may break the ordering invariant. Named and unnamed elements cannot be mixed within one index.

    ix_named <- as_interval_index(
      setNames(list("alice", "bob", "carol"), c("a", "b", "c")),
      start = c(1, 3, 2),
      end = c(4, 5, 6)
    )
    ix_named
    #> Named interval_index with 3 elements, default query bounds [start, end).
    #>
    #> Elements (by interval start order):
    #>
    #> $a (interval 1 - 4)
    #> [1] "alice"
    #>
    #> $c (interval 2 - 6)
    #> [1] "carol"
    #>
    #> $b (interval 3 - 5)
    #> [1] "bob"

    ix_named[["b"]]
    #> [1] "bob"
    ix_named[c("a", "c")]
    #> Named interval_index with 2 elements, default query bounds [start, end).
    #>
    #> Elements (by interval start order):
    #>
    #> $a (interval 1 - 4)
    #> [1] "alice"
    #>
    #> $c (interval 2 - 6)
    #> [1] "carol"
    ix_named[1]
    #> Named interval_index with 1 element, default query bounds [start, end).
    #>
    #> Elements (by interval start order):
    #>
    #> $a (interval 1 - 4)
    #> [1] "alice"

    try(ix_named$a <- "!!")
    #> Error in `$<-.interval_index`(`*tmp*`, a, value = "!!") :
    #>   `$<-` is not supported for interval_index.

## Transforming, iterating, merging

`fapply()` maps a function over elements while preserving intervals and order. The function receives `(value, start, end)`, or `(value, start, end, name)` if it accepts a fourth argument. Endpoints and names are passed in read-only, and the return value replaces the stored element (similar to `lapply()` for named R lists).

    fapply(ix, function(value, start, end) paste0(value, "[", start, ",", end, "]"))
    #> Unnamed interval_index with 3 elements, default query bounds [start, end).
    #>
    #> Elements (by interval start order):
    #>
    #> [[1]] (interval 1 - 3)
    #> [1] "A[1,3]"
    #>
    #> [[2]] (interval 2 - 4)
    #> [1] "B[2,4]"
    #>
    #> [[3]] (interval 4 - 5)
    #> [1] "C[4,5]"

`loop()` (re-exported from the coro package) walks the index in interval start-position order, yielding bare values. Interval endpoints are dropped from each yield; use `fapply()` if your callback needs `(value, start, end)`.

    loop(for (v in ix) print(v))
    #> [1] "A"
    #> [1] "B"
    #> [1] "C"

Plain `for (v in ix)` (without `loop()`) does not dispatch to the iteration protocol, it walks the underlying structure and yields internal nodes rather than elements. Always wrap with `loop()`.

`merge(x, y)` combines two interval indices into a new one in start-position order, preserving left-biased FIFO on tied starts. The general case runs in O(m + n); disjoint start ranges collapse to O(log(min(m, n))) via concat. The reserved `.ivx_min_end` / `.ivx_max_end` monoids recompute automatically, so interval-relation queries work immediately on the result.

    a <- as_interval_index(c("A1", "A2"), start = c(1, 5), end = c(4, 8))
    b <- as_interval_index(c("B1", "B2"), start = c(3, 7), end = c(6, 10))
    m <- merge(a, b)
    peek_all_point(m, 3)
    #> Unnamed interval_index with 2 elements, default query bounds [start, end).
    #>
    #> Elements (by interval start order):
    #>
    #> [[1]] (interval 1 - 4)
    #> [1] "A1"
    #>
    #> [[2]] (interval 3 - 6)
    #> [1] "B1"

Both indices must share the same endpoint type, the same `bounds` convention, and the same monoid set; mismatches error. Both inputs are left unmodified.