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RankedMin

Status: Stable

documented, exercised by the test suite and/or worked examples, with no known limitations recorded.

Description

RankedMin[list, n]

gives the n-th smallest element of list.

RankedMin[list, -n]

gives the n-th largest element of list. RankedMin[list, 1] is Min[list] and RankedMin[list, -1] is Max[list]. Yields a definite result when every element is a real number; +-Infinity are ordered as +-infinity. Has a packed-array fast path and is compilable.

Examples (7)

Every input below was run against the current Mathilda build and its output recorded.

Basic examples (4)

In[1]:= RankedMin[{12, 13, 11}, 2]
Out[1]= 12

In[2]:= RankedMin[{Pi, Sqrt[2], E, 3}, 3]
Out[2]= 3

In[3]:= RankedMax[{2.5, E, 12, 15, 485}, -2]
Out[3]= E

In[4]:= RankedMax[{Infinity, 5, Infinity, -Infinity}, 2]
Out[4]= Infinity

Applications (3)

The 2nd smallest element

In[5]:= RankedMin[{12, 13, 11}, 2]
Out[5]= 12

Symbolic reals order by value

In[6]:= RankedMin[{Pi, Sqrt[2], E, 3}, 3]
Out[6]= 3

The 1st largest is Max

In[7]:= RankedMin[{12, 13, 11}, -1]
Out[7]= 13

Implementation notes

Algorithm. builtin_ranked_min selects the n-th smallest element of a list — RankedMin[list, n], with RankedMin[list, -n] the n-th largest, so RankedMin[list, 1] is Min[list] and RankedMin[list, -1] is Max[list]. It requires an integer, nonzero n, converts it to a 1-based ascending rank r (r = n if positive, m + n + 1 if negative; an out-of-range r or empty list leaves the call unevaluated), and runs ranked_select. One scan chooses the comparison path: if every element is a real numeric atom it compares exactly with expr_compare (BigInt/Rational-safe), otherwise it builds a double key per element with ranked_numeric_key (±HUGE_VAL for ±Infinity, the value of a real atom, else the machine-precision numericalisation of a symbolic real such as Pi, E, Sqrt[2]) — and a non-real element (free symbol, non-real complex) makes it decline, since a definite result needs every element to be a real number. ranked_select_idx then quickselects the r-th position with a Hoare partition and middle pivot (O(m) average), the original index breaking value ties stably, and the exact element at that position is returned.

Data structures. A size_t* idx index array quickselected in place; an optional double* keys for the approximate path; a RankedCtx {Expr** elem; const double* keys;} read by ranked_cmp. Only the r-th slot is guaranteed settled on return. The result is an expr_copy of the selected element in its exact form.

Complexity / limits. O(m) average quickselect. An NDArray argument takes the buffer order-statistic path ndred_ranked_min (int64 exactly, reals via an O(m) quickselect) — RankedMin is on pack.c's AWARE and INT64_OK lists. It is also compilable: CompileDiagnostics[{{v, _Real, 1}}, RankedMin[v, 2]] reports Compiled -> True. Protected.

  • Protected.
  • RankedMax[list, k] is RankedMin[list, -k].
  • RankedMin[list, 1] is Min[list]; RankedMin[list, -1] is Max[list].
  • Yields a definite result whenever every element is a real number, including symbolic real constants (Pi, E, Sqrt[2], Pi + E), which order by value; Infinity/-Infinity rank as ±∞. Returns the element in its exact form.
  • Exact for arbitrary-precision integers and rationals; a symbolic non-real element (a free symbol or a non-real complex), an empty list, or |n| out of range leaves the call unevaluated.
  • Packed-array fast path (int64 exact, real via O(n) quickselect) and a Compile[] lowering, so RankedMin[v, k]/RankedMax[v, k] compile and auto-compile.

Attributes: Protected.

References

See also: RankedMax, Min, Max, Pi, E

Notes & additional examples

Notes

RankedMin[list, n] is the n-th smallest element — the order statistic between Min and Max. A negative index counts from the top, so RankedMin[list, -n] is the n-th largest, RankedMin[list, 1] is Min[list], and RankedMin[list, -1] is Max[list]. It returns a definite result whenever every element is a real number, including symbolic real constants (Pi, E, Sqrt[2]), which order by value, with ±Infinity ranking as ±∞; the element comes back in its exact form. A non-real element, an empty list, or |n| out of range leaves the call unevaluated. Selection is an O(n) quickselect (int64 exact), with a packed-array fast path and a Compile[] lowering.