An entry pairing a hash table key $k$ with its associated value $v$. In the map interface of hash table, each key equality class has at most one live hash table record.

The identifier used to select a hash table record. Key equality determines whether two identifiers name the same entry, while a hash function chooses where its lookup begins.

A hash table collision-handling method storing at most one live record per array slot. A hash function chooses a starting slot, and a probe rule chooses further slots. Unlike separate chaining, records occupy the slots themselves.

Cited rows: Ag I ground configuration and term, atomic weight, and isotope data.

A hadronic form factor parametrizes a current matrix element between hadron states, separating allowed Lorentz covariant tensor structures from scalar-valued functions of Lorentz invariant combinations of momentum transfer. These functions encode composite structure and strong interaction dynamics. Equal kinematic phase space does not imply equal form factors for different weak interaction transitions.

Given a finite set of jobs with fixed half-open intervals $[s_i,f_i)$, where $s_i<f_i$, choose a subset $S$ of maximum cardinality. Each job occupies one shared resource throughout its interval. A feasible solution has pairwise disjoint intervals: for distinct selected jobs $i,j$, either $f_i\le s_j$ or $f_j\le s_i$. Thus a finish and the next start may coincide. Every job has value one; occupied time is not the objective function.^[ref]

Consider these six original jobs, with arbitrary time units:Job $B=[1,3)$ can be followed immediately by $C=[3,5)$ and $D=[5,7)$. Job $A=[0,6)$ conflicts with all three. Starting the resource earlier is therefore not automatically beneficial: one long commitment can displace several shorter requests.

The earliest-finish greedy algorithm uses a sorting algorithm to put jobs in nondecreasing finish time. Scan that order, accepting the first job and then each job whose start is at least the last accepted finish. Break finish-time ties arbitrarily; an empty input returns the empty set. For these jobs, the order and decisions areAfter $B,C$, job $A$ starts before the current finish five and is rejected. Accept $D$; then reject $E=[6,9)$ because six is before seven. Finally accept $F=[8,10)$. The result is $S=\{B,C,D,F\}$, with cardinality four. Leaving $[0,1)$ idle makes room for more jobs.

By contrast, the earliest-start rule scans $A,B,C,D,E,F$. It accepts $A$, skips $B,C,D$, accepts $E$, and skips $F$. Its feasible solution $\{A,E\}$ contains only two jobs. Here $A$ and $E$ meet at time six and are compatible; $E$ and $F$ overlap on $[8,9)$.

There is also an elementary optimality certificate for this example. The four subsets $\{A,B\}$, $\{C\}$, $\{D,E\}$ and $\{F\}$ form a partition of a set of the jobs. A feasible solution takes at most one job from each group: $A,B$ overlap, as do $D,E$, and each other group is a singleton. Hence four is an upper bound, attained by $\{B,C,D,F\}$. Checking all $2^6=64$ subsets gives a unique optimum of size four: $\{B,C,D,F\}$.

The earliest-finish theorem states that this greedy algorithm maximizes job count for every finite instance of interval scheduling. For the exchange argument, let $q$ be the first job in an optimal feasible solution and $g$ an earliest-finishing job. Every later selected job $j$ satisfiesReplace $q$ by $g$; compatibility and cardinality survive. The remainder must be optimal among jobs starting at or after $f_g$, or improving it would improve the whole selection. Mathematical induction on the smaller remaining instance, starting with the empty set, completes the outline.^[ref]

Why is the algorithm fast? Once sorted, each job is examined once, using only the last accepted finish to decide compatibility. The scan takes $O(n)$ time; comparison sorting algorithms give total time complexity $O(n\log n)$ for $n\ge 2$, in Big O notation, assuming constant-cost endpoint comparisons. Already sorted input needs only the $O(n)$ scan.^[ref]

The reason the shortcut is correct is stronger than "finishing sooner looks good": the exchange argument preserves every later job of a suitable optimum. This is the problem-specific fact that permits final decisions without exploring all subsets.